ENGINEERING ECONOMICS

1. An engineer wishes to borrow $20 000 in order to start his own business. A 19. A student plans to deposit $600 each year in a savings account, over a period
bank will lend him the money provided he agrees to repay $920 per month for of 10 years. If the bank pays 6% per year, compounded annually, how much
two years. How much interest is he being charged? money will have accumulated at the end of the 10-year period?
Ans. $2080 Ans. $7908.48
2. A student deposits $1000 in a savings account that pays interest at the rate of 20. Suppose that a fixed sum of money, A, will be deposited in a savings account at
6% per year. How much money will the student have after one year? (b) An the end of each year for 20 years. If the bank pays 6% per year, compounded
investor makes a loan of $5000, to be repaid in one lump sum at the end of one annually, find A such that a total of $50 000 will be accumulated at the end of
year. What annual interest rate corresponds to a lump-sum payment of $5425? the 20-year period.
Ans. (a) $1060; (b) 8.5% Ans. $1359
3. A student borrows $3000 from his uncle in order to finish school. His uncle 21. An engineer who is about to retire has accumulated $50 000 in a savings
agrees to charge him simple interest at the rate of 5½% per year. Suppose the account that pays 6% per year, compounded annually. Suppose that the
student waits two years and then repays the entire loan. How much will he have engineer wishes to withdraw a fixed sum of money at the end of each year for
to repay? 10 years. What is the maximum amount that can be withdrawn?
Ans. $3330 Ans. $6795
4. A student deposits $1000 in a savings account that pays interest at the rate of 22. An engineer who is planning his retirement has decided that he will have to
6% per year, compounded annually. If all of the money is allowed to withdraw $10 000 from his savings account at the end of each year. How much
accumulate, how much will the student have after 12 years? money must the engineer have in the bank at the start of his retirement, if his
Ans. $2012.20 money earns 6% per year, compounded annually, and he is planning a 12-year
5. A student who will inherit $5000 in three years has a savings account that pays retirement (i.e., 12 annual withdrawals)?
5½% per year, compounded annually. What is the present worth of the Ans. $83 839
student's inheritance? 23. An engineer is planning for a 15-year retirement. In order to supplement his
Ans. $4258.07 pension and offset the anticipated effects of inflation, he intends to withdraw
6. An economy is experiencing inflation at the rate of 6% per year. An item $5000 at the end of the first year, and to increase the withdrawal by $1000 at
presently costs $100. If the 6% inflation rate continues, what will be the price of the end of each successive year. How much money must the engineer have in
this item in five years? his savings account at the start of his retirement, if money earns 6% per year,
Ans. $133.82 compounded annually?
7. An economy is experiencing inflation at an annual rate of 6%. If this continues, Ans. $106 116.59
what will $100 be worth five years from now, in terms of today's dollars? 24. Find the present value, in pesos, of an annuity of 25,000 pesos payable
Ans. $74.73 annually for 8 years, with the first payment at the end of 10 years, if money is
8. The ABC Company deposited $100 000 in a bank account on June 15 and worth 5%.
withdrew a total of $115 000 exactly one year later. Compute: (a) the interest Ans. P104,156
which the ABC Company received from the $100 000 investment, and (b) the 25. A nominal annual rate of interest of 14%, compounded continuously, has an
annual interest rate which the ABC Company was paid. effective annual interest rate of:
Ans. (a) $15,000 (b) 15% Ans. 15.03%
9. What is the annual rate of simple interest if $265 is earned in four months on an 26. A man receives P145,000.00 credit for his old car when buying a new model
investment of $15000? costing P375,000.00. What cash payment will be necessary so that the balance
Ans. 5.3% can be liquidated by payments of P12,500.00 at the end of each month for 18
10. Determine the principal that would have to be invested to provide $200 of months when interest is charged at the rate of 6% compounded monthly?
simple interest income at the end of two years if the annual interest rate is 9% Ans. 15,340.00
Ans. $1111.11 27. What is the smallest nominal rate to yield an effective rate of 10.15%?
11. Compare the interest earned from an investment of $1000 for 15 years at 10% Ans. 9.667%
per annum simple interest, with the amount of interest that could be earned if 28. Find the value after 20 years, in pesos, of an annuity of P20,000 pesos payable
these funds were invested for 15 years at 10% per year, compounded annually. annually for 8 years with the first payment at the end of 2 years, if money is
Ans. $3177.25 worth 5%.
12. At what annual interest rate is $500 one year ago equivalent to $600 today? Ans. P326,644
Ans. 20% 29. A machine is to be purchased for P155,000; it has an estimated life of 8 years
13. Suppose that the interest rate is 10% per year, compounded annually. What is and a salvage value of P6,000. A sinking fund is to be established so money will
the minimum amount of money that would have to be invested for a two-year be available to purchase a replacement when the first machine wears out at the
period in order to earn $300 in interest? end of 8 years. An amount of P13,030 is to be deposited at the end of each year
Ans. $1428.57 during the lifetime of the first machine into the sinking fund. The interest rate (%)
14. How long would it take for an investor to double his money at 10% interest per this fund must earn to produce sufficient fund to purchase the replacement
year, compounded annually? machine at the end of eight years is closest to:
Ans. 8 years Ans. 10%
15. Suppose that a man lends $1000 for four years at 12% per year simple interest. 30. An employee is earning P18,000 a month and he can only afford to purchase a
At the end of the four years, he invests the entire amount which he then has for car which will require a down payment of P85,000 and a monthly amortization of
10 years at 8% interest per year, compounded annually. How much money will 30% of his monthly salary. What would be the maximum cash value of a car he
he have at the end of the 14-year period? can purchase if the seller will agree to a down payment of P85,000 and the
Ans. $3195.21 balance payable in four years at 18% per year payable in monthly basis. The
16. Let the inflation rate be 6% per year. If a person deposits $50 000 in a bank first payment will be due at the end of the first month.
account at 9% per annum simple interest for 10 years, will this effectively Ans. P268,830
protect the purchasing power of the original principal? 31. An equipment is bought at P420,000 with an economic life of 6 years and a
Ans. $53,047.50 > $50,000. Thus, the purchasing power of the investment will salvage value of P50,000. The first year depreciation is P125,420. The cost of
be protected, and a small amount of interest will be earned money is 12% per year. What method of depreciation was used?
17. An individual wants to have $2000 at the end of three years. How much would Ans. declining balance method C. SOYD method
the individual have to invest at a 10% per year interest rate, compounded B. sinking-fund method D. straight-line method
annually, in order to obtain a net of $2000 after paying a $250 early withdrawal 32. A cigarette vendor borrowed P2,400 and agreed to pay P3,000 after 30 days.
fee at the end of the third year? What is the simple interest rate per annum?
Ans. $1690.46 Ans. 300%
18. A certain sum of money will be deposited in a savings account that pays interest 33. With interest rate of 9% compounded continuously, what is the present worth of
at the rate of 6% per year, compounded annually. If all of the money is allowed perpetuity of P8,000 payable monthly?
to accumulate, how much must be deposited initially so that $5000 will have Ans. P1,062,699
accumulated after 10 years?
Ans. $2792.00
ENGINEERING ECONOMICS
34. An investor pays P1,100,000 for a mine which will yield a net income of x pesos 49. A woman deposits $2000 in a savings account that pays interest at 8% per
at the end of each year for 10 years and then will become valueless. He year, compounded annually. If all the money is allowed to accumulate, how
accumulates a replacement fund to recover his capital by annual investments at much will she have at the end of (a) 10 years? (b) 15 years?
4.5%. Find x if he desires 11.5% return on his investment. Ans.(a) $4317.80 (b) $6344.40
Ans. P216,000 50. How much money must be deposited in a savings account so that $5500 can be
35. In what price will you sell a cell phone for sale that costs P6000 in order that you withdrawn 12 years hence, if the interest rate is 9% per year, compounded
may offer 20% discount on the selling price and still make a profit of 25% on the annually, and if all the interest is allowed to accumulate?
selling price? Ans. $1955.42
Ans. 10,000 51. Suppose that a person deposits $500 in a savings account at the end of each
36. An electric replacement pump is being considered for purchase. It is capable of year, starting now, for the next 12 years. If the bank pays 8% per year,
providing 200 hp. The pertinent data are as follows: compounded annually, how much money will accumulate by the end of the 12-
Cost = P3,200 Maintenance cost per year = P50 year period?
Electric efficiency = 0.85 Life expectancy = 14 yrs. Ans. $9488.55
The pump is used for 400 hours per year and the cost of electricity is P0.04 per 52. How much money must be deposited at the end of each year in a savings
kilowatt-hour, (1 horsepower = 0.746 kW). Assuming the pump will have no account that pays 9% per year, compounded annually, in order to have a total of
salvage value and using Straight line depreciation, what will be the monthly $10 000 at the end of 14 years?
cost? Ans. $384.33
Ans. 238.20 53. A bank advertises that it pays interest at the rate of 10% per year, compounded
37. Terms of sales: P90,000 payable in 120 days or P85,500 payable in 45 days. quarterly. What effective interest rate is the bank paying?
Find the equivalent annual rate of simple interest if paid in 120 days. Ans. 2.5% per quarter
Ans. 28.24% 54. Mr. Smith is planning his retirement. He has decided that he needs to withdraw
38. A man wants to make 14% nominal interest compounded semi-annually on a $12000 per year from his bank account to supplement his other income from
bond investment. How much should the man be willing to pay now for a 12%, Social Security and a private pension plan. How much money should he plan to
P40,000 bond that will mature in 10 years and pays interest semi-annually? have in the bank at the start of his retirement, if the bank pays 10% per year,
Ans. P35,762 compounded annually, and if he wants money to last for a 12-year retirement
39. A man has deposited $50 000 in a retirement income plan with a local bank. period?
This bank pays 9% per year, compounded annually, on such deposits. What is Ans. $81 766.15
the maximum amount the man can withdraw at the end of each year and still 55. Mr. Doe is trying to decide whether to put his money in the XYZ Bank or the
have the funds last for 12 years? ABC Bank. The XYZ Bank pays 6% per annum interest, compounded annually;
Ans. $6982.50 the ABC Bank pays 5% per annum interest, compounded quarterly. Mr. Doe
40. Peter borrows P10,000 from a bank on the first days of the month. The interest expects to keep his money in the bank for 5 years. Which bank should he
is computed at the end of every month on the amount he still owed at the rate of select?
1.5% per month. Peter pays “x” pesos at the end of the first month, another “x” Ans. for XYZ: (F/P, 6%,5) = 1.3382. for ABC: (F/P, 1¼:%, 20) = 1.2820; He
at the end of the second month and another “x” at the end of the 3rd month. should choose XYZ, which offers the greater return per dollar.
Peter has completely paid off the debt completely at the end of the 3rd month, 56. Mr. Franklin wants to save for a new sports car that he expects will cost $38 000
find the value of “x”. four and one-half years from now. How much money will he have to save each
Ans. 3434 year and deposit in a savings account that pays 6¼% per year, compounded
41. A man bought a government bond which cost P1000 and will pay P50 interest annually, to buy the car in four and one-half years?
each year for 20 yrs. The bond will mature at the end of 20 years and he will Ans. $8654.32 per year for four years
receive the original P1000. If there is 2% annual inflation during this period, 57. Dr. Anderson plans to make a series of gradient-type withdrawals from her
what rate of return will the investor receive after considering the effect of savings account over a 10-year period, beginning at the end of the second year.
inflation? What equal annual withdrawals would be equivalent to a withdrawal of $1000 at
Ans. 2.94% the end of the second year, $2000 at the end of the third year,. . . , $9000 at the
42. Compute the interest for an amount of P200,000 for a period of 8 yrs. if it was end of the 10th year, if the bank pays 9% per year, compounded annually?
made at 16% compounded continuously. Ans. $3797.80
Ans. 519,328 58. Mr. Jones is planning a 20-year retirement; he wants to withdraw $6000 at the
43. A machine costs P900,000 and will have a salvage value of P450,000 when end of the first year, and then to increase the withdrawals by $800 each year to
retired at the end of 5 yrs. Using the sum of years digit method, what is the sum offset inflation. How much money should he have in his savings account at the
of the depreciation cost in the first two years. start of his retirement, if the bank pays 9% per year, compounded annually, on
Ans. P270,000 his savings?
44. What rate of interest compounded monthly is equivalent to an interest rate of Ans. $104 189.14
14% compounded quarterly? 59. The ABD Company is building a new plant, whose equipment maintenance
Ans. 13.84% costs are expected to be $500 the first year, $150 the second year, $200 the
45. A man wished to have P40,000 in a certain fund at the end of 8 years that will third year, $250 the fourth year, etc., increasing by $50 per year through the
pay a nominal rate of 6% compounded continuously. What is the value of the 10th year. The plant is expected to have a 10-year life. Assuming the interest
compound amount factor for this rate? rate is 8O/0, compounded annually, how much should the company plan to set
Ans. 1.616 aside now in order to pay for the maintenance?
46. An equipment installation job of Diego Construction in the completion stage can Ans. $2340.20
be completed in 40 days of 8 hrs. per day of work with 40 men working. With the 60. Mr. Holzman estimates that the maintenance cost of a new car will be $75 the
contract expiring in 30 days, the contractor decided to add 10 men on the job, first year, and will increase by $50 each subsequent year. He plans to keep the
overtime not being permitted. If the liquidated damages P20,000 per day of car for 6 years. He wants to know how much money to deposit in a bank
delay, and the men are paid P580 per day, compute the total cost if he will add account at the time he purchases the car, in order to cover these maintenance
10 more men to finish the job. costs. His bank pays 5% per year, compounded annually, on savings deposits.
Ans. P968,000 Ans. $960.17
47. In 1940, the average value of a house is P290,000. In 1990, the average value 61. How many years will be required for a given sum of money to triple, if it is
of a house of the same model is P7,910,000. What was the rate of inflation for deposited in a bank account that pays 6% per year, compounded annually?
the house? Ans. 19 years
Ans. 6.8% 62. Determine the value of n corresponding to A/F = 0.01, if i = 7½% per year,
48. An equipment costs P325,000 and has a life of 4 yrs. With a salvage value of compounded annually.
P50,000. Determine the capitalized cost of using the machine if the rate of Ans. 30
interest is 16% per annum. 63. A bank will return $2345 on a 10-year certificate of deposit that originally cost
Ans. P664,238 $1000. What interest rate, compounded annually, is the bank paying?
Ans. 8.89%
ENGINEERING ECONOMICS
64. A bank claims to pay interest to its depositors at the rate of 6% per year, 80. The first cost of a certain equipment is P324000 and a salvage value of P50,000
compounded quarterly. What are the nominal and effective interest rates? at the end of its life for 4 yrs. Money is worth 6% annually. If there is no
Ans. 1.5% per quarter salvage value and the annual maintenance cost is P18000, find the capitalized
65. An engineer plans to borrow $3000 from his company credit union, to be repaid cost of perpetual service.
in 24 equal monthly instalments. The credit union charges interest at the rate of Ans. 624,000
1% per month on the unpaid balance. How much money must the engineer 81. A machine costs P100,000 with a salvage value of P20,000 at the end of its life
repay each month? of 8 years. Which of the following gives the depreciation at the 6th year using
Ans. $141.21 per month sinking fund method if money is worth 6% annually?
66. An engineer wishes to purchase an $80 000 home by making a down payment Ans. 8082.88
of $20 000 and borrowing the remaining $60 000, which he will repay on a 82. A building has a salvage value of 1 million after 50 years. Annual depreciation is
monthly basis over the next 30 years. If the bank charges interest at the rate of 2 million. Using the straight-line method of depreciation. If the building is to be
9½% per year, compounded monthly, how much money must the engineer sold for 30 million, how many years after should you sell it?
repay each month? Ans. 35.5 years
Ans. $504.51 83. A machine was purchased at an original cost of P400,000.00 with a salvage
67. An engineer deposits $1000 in a savings account at the end of each year. If the value of P20,000.00. Life of this machine is expected to last for 6 years. It was
bank pays interest at the rate of 6% per year, compounded quarterly, how much used for 4000 hrs. in the first year, 6000 hrs. in the second year and 8000 hrs.
money will have accumulated in the account after 5 years? on the third year. The machine is expected to last for 3800 hours in the period
Ans. $5652.50 of 6 years. Which of the following gives the depreciation at the end of the
68. A bank pays interest at the rate of 6% per year, compounded monthly. If a second year?
person deposits $2500 in a savings account at the bank, how much money will Ans. 60,000
accumulate by the end of 2 years? 84. A local company assembling stereo radio cassette produces 300 units per
Ans. $2817.89 month at a cost of P800.00 per unit. Each stereo radio cassette sells for
69. A man plans to buy a $150 000 house. He wants to make a down payment of P1,200.00. If the firm makes a profit of 10% on its, 10,000 shares with a par
$30 000 and to take out a 30-year mortgage for the remaining $120 000, at 10% value of P200.00 per share, and the total fixed cost are P20,000 per month.
per year, compounded monthly. How much must he repay each month? Which of the following gives the break-even point?
Ans. $1053.08 Ans. 92 units
70. A man plans to save $1000 a month for the next 20 years, at 10% per year, 85. The Asian Transmission Co. makes and sells certain automotive parts. Present
compounded monthly. How much money will he have at the end of 20 years? sales volume is 500,000 units a year at a selling price of fifty centavos (P0.50)
Ans. $759,371.43 per unit. Fixed expenses total P80,000.00 per year. Which of the following
71. What is the present value of a stream of monthly payments of $500 each over gives the present total profit for a year?
10 years, if the interest rate is 10% per annum, compounded monthly? Ans. 170,000
Ans. $37 835.72 86. A proposed manufacturing plant will require a fixed capital investment of P8 M
72. How much money must be deposited in a savings account each month to and an estimated working capital of P1.5 M. The annual profit is P2 M and the
accumulate $10 000 at the end of 5 years, if the bank pays interest at the rate of annual depreciation is estimated to be 8% of the fixed capital investment.
6% per year, compounded (a) monthly? (b) semi-annually? (c) quarterly? (d) Compute the rate of return of the total investment.
daily? Ans. 14.32%
Ans. (a) 143.33 per month (b) $872.30 every 6 months or 145.38 per month (c) 87. A fixed capital investment of P10,000,000 is required for a proposed
$432.45 per quarter (d) $143.28 per month manufacturing plant and an estimated working capital of P2,000,000. Annual
73. Mrs. Carter deposits $100 in the bank at the end of each month. If the bank depreciation is estimated to be 10% of the fixed capital investment. Which of the
pays (a) 6% per year, (b) 7% per year, compounded monthly, how much money following gives the payout period in years?
will she have accumulated at the end of 5 years? Ans. 2.86 yrs.
Ans. (a) 6977.00 (b) 7159.29 88. A coin making machine costing P200,000 has a salvage value of P20,000 at the
74. An investment of P8.5 million is expected to yield an annual income of P2.8 end of its economic life of 5 yrs. The schedule of production per year is as
million. Determine the recovery period in yrs. based on the following estimates: follows:
Annual depreciation = P 1.0M Year Number of Coins
Operational expenses = P 0.6M 1 100,000
Taxes and insurance = P 0.2M 2 80,000
Miscellaneous expenses = P50,000 3 60,000
Ans. 8.9 4 40,000
75. A broadcasting corporation was formed duly approved by the Securities and 5 20,000
Exchange office has a working capital of P20 M and a fixed capital of P80 M. Determine the annual reserve for depreciation for the 3rd year only.
Annual depreciation amounts to P5 M and expected annual profit is P16 M. Ans. 36,000
Determine the recovery period in years. 89. A construction equipment is badly needed for a certain project so as to shorten
Ans. 6.25 the time of completion of the project. The equipment cost P1.2 million and has
76. Compute the equivalent nominal annual interest compounded continuously if a life of 5 yrs. with a salvage value of P200,000 at the end of its life. This
the effective annual interest rate is 4%. machine can be bought with money borrowed at an interest rate of 20% per
Ans. 3.92% annum. Annual operating cost is P10,000. Determine the capitalized cost of
77. The cost of equipment is P500,000 and the cost of installation, labor, taxes and using this equipment.
miscellaneous expenses is P30000. If the salvage value is 10% of the cost of Ans. 1,921,899
equipment at the end of its life of 5 years, compute the book value at the end of 90. A fixed capital investment of P10,000,000 is required for a proposed
the 3rd year using MARCS Method. manufacturing plant and an estimated working capital of P2,000,000. Annual
Ans. 152,640 depreciation is estimated to be 10% of the fixed capital investment. Which of the
78. A telephone company purchased a microwave radio equipment for P6M. If the following gives the recovery period in years?
equipment shall be depreciated over a period of 8 yrs. with a salvage value of A. 4.8 yrs.
5% of the cost of the equipment, determine the worth of the equipment on the 91. A textile mill has just purchased a lift truck that has a useful life of 5 years. The
4th year after deducting depreciation charges using MARCS Method. engineer estimates that the maintenance costs for the truck during the first year
Ans. 2.215 M will be $1000. Maintenance costs are expected to increase as the truck ages at
79. A contractor can buy dump trucks for P800,000 each (surplus) or rent them for a rate of $250 per year over the remaining life. Assume that the maintenance
P1189 per truck per day. The truck has a salvage value of P100,000 at the end costs occur at the end of each year. The firm wants to set up a maintenance
of its useful life of 5 yrs. Annual cost of maintenance is P20,000.00. If money is account that earns 12% annual interest. All future maintenance expenses will be
worth 14% per annum. Determine the number of days per year that a truck must paid out from this account. How much does the firm have to deposit in the
be used to warrant the purchase of the truck. account now?
Ans. 200 days Ans. $5204
ENGINEERING ECONOMICS
92. An engineering school has just completed a new engineering complex worth 107. Ms. Brown deposits $1000 in the bank at the end of the first year, $1200 at the
$50 million dollars. A campaign targeting alumni, is planned to raise funds for end of the second year, etc., continuing to increase the amount by $200 a year,
the future maintenance costs, estimated at $2 million per year. Any unforeseen for 20 years. If the bank pays 7% per year, compounded continuously, how
above $2 million per year would be obtained by raising tuition. Assuming that much money will have accumulated at the end of 20 years?
the school can create a trust fund that earns 8% interest annually, how much Ans. $103 193.35
has to be raised now to cover the perpetual string of $2 million annual costs? 108. Mr. Smith is planning his retirement. He has decided that he will need $12 000
Ans. $25,000,000 per year to live on, in addition to his other retirement income from Social
93. A bridge just constructed costs $1,000,000. It is estimated that the same bridge Security and a private pension plan. How much money should he plan to have
should be renovated every 50 years at a cost of $800,000. Annual repairs and in the bank at the start of his retirement, if the bank pays 10% per year,
maintenance are estimated to be $30,000 per year. If interest rate is 5%, compounded continuously, and if Mr. Smith wants to make 12 annual
determine the capitalized cost of the bridge. withdrawals of $12 000 each?
Ans. $1,676,428 Ans. $79 734.22
94. To decrease costs of operating a lock in a large river, a new system of operation 109. Mrs. Carter deposits $100 in the bank at the end of each month. If the bank
is proposed. It will cost $450,000 to design and build. It is estimated that it will pays (a) 6% per year, (b) 7% per year, compounded continuously, how much
have to be reworked every 10 years at a cost of $50,000. In addition, there will money will she have accumulated at the end of 5 years?
be an expenditure of $40,000 at the end of the fifth year for a new type of gear Ans. (a) $6979.70 (b) $7163.56
that will not be available until then. The annual operating costs are expected to 110. Mrs. Carter deposits $100 a month during the first year, $110 a month during
be $30,000 for the first 15 years and $25,000 a year thereafter. Compute the the second year, $120 a month during the third year, etc. How much will have
capitalized cost of perpetual service at i = 10%. accumulated at the end of 5 years if the interest rate is 6% per year,
Ans. $793,708 compounded continuously?
95. Consider the following data on an automobile: Ans. $8291.87
Cost of asset, I = $10,000 111. Suppose that $2000 is deposited each year, on a continuous basis, into a
Useful life, N = 5 years savings account that pays 6% per year, compounded continuously. How much
Estimated salvage value, S = $2,000 money will have accumulated after 12 years?
Compute the annual depreciation allowance using the straight-line depreciation Ans. $35 147.77
method. 112. ABC Corporation has decided to sell $1000 bonds which will pay semi-annual
Ans. $1,600 dividends of $20 (2% per period) and will mature in 5 years. The bonds are sold
96. A truck for hauling coal has an estimated net cost of $50,000 and is expected to at $830, but after brokers' fees and other expenses the company ends up
give service for 250,000 miles, resulting in a zero-salvage value. Compute the receiving $760. What is the company's cost of the capital raised through the
allowed depreciation amount for the truck usage of 30,000 miles. sale of these bonds?
Ans. $6,000 Ans. 10.62%
97. Mrs. Carter deposits $100 a month during the first year, $110 a month during 113. An engineering firm has turned to Friendly Shark, Inc., to borrow $30000
the second year, $120 a month during the third year, etc. How much will have needed for a short-term (2-year) project, attracted by an advertisement
accumulated at the end of 5 years if the interest rate is 6% per year, announcing an interest rate of 12% per year. Friendly Shark's loan statement
compounded monthly? indicates the following:
Ans. $8289.18 Interest: ($30 000) (1% per month) (24 months) = $7 200
98. A savings account earns interest at the rate of 6% per year, compounded Loan 30 000
continuously. How much money must initially be placed in the account to Total $37 200
provide for twenty end-of-year withdrawals, if the first withdrawal is $1000 and Monthly instalment = $37 200/24 = $1550
each subsequent withdrawal increases by $200? What is the actual cost of borrowing money from Friendly Shark, Inc.?
Ans. $28 368.42 Ans. 23.87%
99. A person borrows $5000 for 3 years, to be repaid in 36equal monthly 114. A house is being advertised for sale by the owner. An investor estimates that
instalments. The interest rate is 10% per year, compounded continuously. How the property could be rented out for $600 per month. Taxes and minor
much money must be repaid at the end of each month? maintenance expenses are estimated at $1200 per year. The house has been
Ans. $161.43 recently remodelled, and the tenant should have to pay all utilities. The investor
100. A bank offers its customers a Christmas Club account, in which they deposit thinks he could sell the house for $85 000 after 5 years. What is the largest
$12.61 a week for 39 weeks, starting in mid-February. At the end of 39 weeks amount that the investor can offer for the property if his MARR is 12%,
(mid-November), each customer will have accumulated $500, which can be compounded monthly?
withdrawn to pay for gifts and other seasonal expenses. What is the nominal Ans. $69 511.62
interest rate, assuming continuous compounding? 115. If the interest rate is 8% per year, compounded annually, what is the equivalent
Ans. 4.52% present value of $10 000 (a) 1 year from today? (b) 5 years from today?
101. At what rate must funds be continuously added to a savings account in order to Ans. (a) $9259.26 (b) $6805.96
accumulate $10 000 in 15 years, if interest is paid at 5% per year, compounded 116. What is the equivalent future value of $1000 annually for the next 9 years, if the
continuously? interest rate is 8% per year, compounded annually?
Ans. $447.63 per year Ans. $12 487.60
102. What effective annual interest rate corresponds to a nominal interest rate of 117. What amount of money is equivalent to receiving $8000 three years from today,
10% per year, compounded continuously? if the interest rate is 8% per year, compounded semi-annually?
Ans. 10.52% Ans. $5632.46
103. Determine the nominal interest rate corresponding to an effective interest rate of 118. What is the equivalent present value of the following series of payments: $5000
10% per year, compounded continuously. the first year, $5500 the second year, and $6000 the third year? The interest
Ans. 0.0953 rate is 8%, compounded annually.
104. How much money must be deposited in a savings account so that $5500 can be Ans. $14 108.06
withdrawn 12 years hence, if the interest rate is 9% per year, compounded 119. What single amount at the end of the fourth year is equivalent to a uniform
continuously, and if all the interest is allowed to accumulate? annual series of $3000 per year for 10 years, if the interest rate is 10% per year,
Ans. $1867.76 compounded annually?
105. How much money must be deposited at the end of each year in a savings Ans. $26 988.02
account that pays 9% per year, compounded continuously, to have a total of 120. A series of 10 annual payments of $2000 is equivalent to two equal payments,
$10 000 at the end of 14 years? one at the end of 15 years and the other at the end of 20 years. The interest
Ans. $372.90 rate is 8%, compounded annually. What is the amount of the two equal
106. A bank pays 6% interest per year, compounded (a) quarterly, (b) annually. A payments?
$5000 deposit will grow to what amount if left in that bank for 2 years? Ans. $25 331.08
Ans. (a) $5,632.46 (b) 5618.00
ENGINEERING ECONOMICS
GEOMETRY PRACTICE PROBLEMS
1. Each interior angle of a regular polygon with nine sides is: cylindrical reservoir is 3.6 m in diameter and 10 m deep is emptied to the pool. Find the
Ans. 140° depth of water at the deep end.
2. Find the sum of the interior angles of a pentagon. Ans. 2.842 m
Ans. 540° 26. The area of a lune is 100 sq. cm. If the area of the sphere is 720 sq. cm, what is the angle
3. How many sides has a polygon if the sum of its interior angles is thrice the sum of its exterior of the lune?
angles? Ans. 50
Ans. 8 27. A light bulb is placed at a certain distance from the surface of a spherical globe of radius 20
4. A road is tangent to a circular lake. Along the road and 12 miles from the point of tangency, cm. If it illuminates one-third of the total surface of the globe, how far it is from the surface?
another road opens towards the lake. From the intersection of the two roads to the periphery Ans. 40 cm
of the lake, the length of the new road is 11 miles. If the new road will be prolonged across 28. A sphere of radius 5 cm and a right circular cone of radius 5 cm and height 10 cm stand on
the lake, find the length of the bridge to be constructed. a plane. How far from the base of a cone must cutting a plane (parallel to the base of the
Ans. 2.091 mi cone) pass in order to cut the solids in equal circular sections?
5. A circle is inscribed and circumscribed about another. Determine the ratio of the area of Ans. 2 cm
larger square to the area of the smaller square. 29. An ice cream cone is filled with ice cream and more ice cream in the form of a hemisphere
Ans. 2:1 is place on top. The diameter of the hemisphere is equal to the diameter of the cone. If the
6. The parallel sides of a trapezoidal lot of measure 160 m and 240 m and are 40 m apart. Find hemispherical surface is equal to lateral area of the cone, find the total volume of ice cream
the length of the dividing line parallel to the two sides that will divide the lot into two equal if the radius of the hemisphere is 2 cm.
areas. Ans. 31.3 cc
Ans. 203.96 30. The volume of the frustum of a regular triangular pyramid is 15 times the edge of the lower
7. A square section ABCD has one of its sides equal to a. Point E is inside the square forming base. The lower base is an equilateral triangle with an edge of 9 m. The upper base is 8 m.
an equilateral triangle BEC having one side equal in length to the side of the square. Find above the lower base. What is the upper base edge?
the angle AED. Ans. 3
Ans. 150° 31. A spherical sector is cut from a sphere whose diameter is 24 cm. Find its volume if its central
8. Points A, B, C, D and E is on the periphery of the circle. AB = BC = CD. If the value of the angle is 30.
angle BAC is 350°, find the angle DEA. Ans. 123.3
Ans. 105° 32. The capacities of two hemispherical tanks are in the ratio of 64:125. If 4.8 kg of paint is
9. A circle having a radius of 4 cm. is inscribed in a square section. A smaller circle is also required to paint the outer surface of the smaller tank, then how many kg of paint will be
tangent to the two sides of the square and to the bigger circle, which is inscribe in a square. needed to paint the outer surface of the larger tank?
Compute the radius of the smaller circle. Ans. 7.5
Ans. 1.48 33. Three spheres made of lead have a radius of r, 2r, and 4r, respectively are melted to form a
10. A trapezium ABCD has smaller side AB parallel to the bigger side CD. Side AD makes an new sphere of radius R. The ratio of the volume to the surface area of the new sphere is
angle of 45° from the side CD while side BC makes an angle of 60° with side CD. If AD = 1 equal to 418. Compute the radius “r”.
cm, compute the value of the side BC. Ans. 3
Ans. 0.816 34. A right circular cone is divided into 3 portions A, B, and C by planes parallel to the base. The
11. A swimming pool has the form of two intersecting circles of equal radius of 30 m. If the center height of each portion is h and the base radius of the cone A is r units. Cone A is at the
of each circle lies on the circumference of the other, compute the perimeter of the swimming vertex of the big cone. Determine the ratio of the volume of A to that of B.
pool. Ans. 1:7
Ans. 215.33 m 35. By using Pappus Theorem, determine the volume generated by revolving the area in the
12. PA and PB are tangents at A and B respectively of a circle having a diameter AC. If AC and first and second quadrants bounded by the ellipse 4x² + 25y² = 100 and the x-axis , about
PB is prolonged, it will intersect outside the circle at Q. If the value of the angle PQA is 20°, the x-axis.
find the value of the angle BAQ. Ans. 83.78
Ans. 35° 36. Find the volume generated by revolving the triangle whose vertices are (2,2), (4,8), and (6,2)
13. Secants PB and PD are drawn from point P having an angle between them equal to 20°. about the line 3x – 4y = 12.
The secant intersects a circle at points C, B, D, and A. C is along the secant PB while A is Ans. 241
along the secant PD, with C and A nearer to P. The angle PBA = 40°. If AB is a diameter of 37. The intercept of a straight line on the y-axis is -3. If (5, 2) is a point on the straight line, the
the circle, find the angle ACD. slope of the straight line is:
Ans. 30° Ans. 1
14. A quadrilateral ABCD is inscribed in a circle having a diameter AD = 8.224 cm. If AB = 2 cm, 38. Line AB has point A(4, 5) and point B(-3, -2). Find point C along line AB if distance BC is
BC = 4 cm, and CD = 6 cm. Find the area of the quadrilateral. three times the distance AC.
Ans. 19.6 Ans. (2.25, 3.25)
15. The volume of a spherical wedge is 6.67 cubic meters and its radius is 2 m. Determine the 39. If the angle from the line 2x + 5y – 17 = 0 to the line 3x – By – 8 = 0 is 45°, determine the
surface area of the lune. value of B.
Ans. 10 Ans. 7
16. The lateral area of a right circular cone is 634 square meters. If its diameter is two-thirds its 40. If the curve Ax² + By² + F = 0 passes through (0, 3) and (3, 0), the curve is:
altitude, determine its altitude in meters. Ans. a circle
Ans. 25.26 41. What is the total length of the curves x² + y² + 4x – 10y - 92 = 0?
17. The surface area of a regular tetrahedron is 173.2 square centimeters. What is its volume? Ans. 69.12
Ans. 117.85 cc 42. The distance between points (7, 3, 6) and (2, 10, 4) is:
18. A cube of edge 30 cm is cut by a plane containing two diagonally opposite edges of the Ans. 8.83
cube. Find the area of the section thus formed in sq. cm. 43. Solve for y if point (6, y) is equidistant to points (3, 6), (8, 12)
Ans. 1273 Ans. 8.583
19. The bases of a right prism are pentagons with each side 6 cm long. The bases are 14 cm 44. Find the focus and directrix of the parabola y = -⅙x².
apart. What is the volume of the prism in cu. cm? Ans. F(0, -3/2); directrix: y = 3/2
Ans. 867 45. (a) Find an equation of a parabola that has vertex at the origin, opens right, and passes
20. A solid pyramid whose altitude is 1.5 m weighs 2000 kg. At what distance from its base must through the point P(7, -3). (b) Find the focus of the parabola.
it be cut by a plane parallel to its base so that two solids of equal weight will be formed. 7
Ans. (a) x = y² (b) (9/28, 0)
Ans. 0.31 m 9
46. A parabola has vertex V(-4, 2) and directrix y = 5. Express the equation of the parabola in
21. A pit 5 m deep was dug out from the ground. The pit is 3 m by 4 m at the top and 2 m by 3
the form y = ax² + bx + c.
m at the bottom. What is the volume of earth removed in cubic meter? 1 2 2
Ans. 𝑦 = − 12 𝑥 2 − 3 𝑥 + 3
Ans. 1212
22. A solid spherical steel ball x cm in diameter is placed into a tall vertical cylinder 12 cm in 47. The interior of a satellite TV antenna is a dish having the shape of a (finite) paraboloid that
diameter containing water, causing the water level to rise by 6 cm. What is the value of x? has diameter 12 feet and is 2 feet deep. Find the distance from the center of the dish to the
Ans. 10.9 cm focus.
23. A trough, whose ends are isosceles right triangles with vertical axis, is 20 ft long. If it contains Ans. 4.5 feet
40 gallons of water, how deep is the water? (1 gallon = 231 cubic inches) 48. Sketch the graph of 2x² + 9y² = 18, and find the foci.
Ans. 6.2 in Ans. F(√7, 0) and F’(-√7, 0)
24. A cross-section of a trough is a semi-ellipse with width at the top 18 cm and depth 12 cm. 49. Sketch the graph of 9x² + 4y² = 25, and find the foci.
The through is filled with water to a depth of 8 cm. Find the width of the surface of the water. Ans. F(0, 1.86) and F’(0, -1.86)
Ans. 16.97 cm 50. Find an equation of the ellipse with vertices (±4, 0) and foci (±2, 0).
25. A swimming pool is rectangular in shape length 12 m and width 5.5 m. It has a sloping bottom Ans. x²/16 + y²/12 = 1
and is 1 m deep at one end and 3.6 m deep at the other end. The water from the full
GEOMETRY PRACTICE PROBLEMS
51. Halley’s comet has an elliptical orbit with eccentricity e = 0.967. The closest that Halley’s 76. A circle has a radius of 5 with its center at O. Find the equation of the tangent to the circle
comet comes to the sun is 0.587 AU. Approximate the maximum distance of the comet from at (3, 4).
the sun, to the nearest 0.1 AU. Ans. 3x + 4y = 25
Ans. 35.0 AU. 77. A circle has an equation of x² + y² + 2ky = 0. Find the value of k when the length of the
52. Sketch the graph of 9x² - 4y² = 36. Find the foci and equations of the asymptotes. tangent from (5,4) to the circle is equal to one.
Ans. F(√13, 0) and F’(-√13, 0), y = ± x
3
Ans. - 5
2
78. A parabola has an equation of x² = 20y. Determine the equation of the directrix of the
53. A hyperbola has vertices (±3, 0) and passes through the point P(5, 2). Find its equation and
asymptotes. parabola.
Ans. y + 5 = 0
Ans. x² - 4y² = 9
79. A curve has an equation of x² + 16y² – 16x + 96y + 144 = 0. Find the equation of the tangent
54. Coast Guard station A is 200 miles directly east of another station B. A ship is sailing on a
to the curve at point (8, - 1).
line parallel to and 50 miles north of the line through A and B. Radio signals are sent out
from A and B at the rate of 980 ft/μsec (microsecond). If, at 1:00 P.M., the signal from B A. y + 1 = 0
80. The points (1, 2, 6) and (1, 6, 2) are vertices of an equilateral triangle. The other vertex is at
reaches the ship 400 microseconds after the signal from A, locate the position of the ship at
(5, 2, z). Solve for z.
that time.
Ans. 2
Ans. At coordinates (42, 50)
55. The path of a projectile at time t can be modeled using the parametric equations 81. Find the latus rectum of the curve r Sin2 = Cos .
1 Ans. 1
𝑠(𝑡) = (𝑠 cos 𝛼)𝑡, 𝑦(𝑡) = − 𝑔𝑡 2 + (𝑠 sin 𝛼)𝑡 + ℎ; 𝑡 ≥ 0, 82. The vertices of a triangle have polar coordinates of (0, 10°), (6, 30°), and (9, 70°). Find the
2
where, at t = 0, s is the speed of the projectile in ft/sec, 𝛼 is the angle the path makes with area of the triangle.
the horizontal, and h is the height in feet. The acceleration due to gravity is g = 32 ft/sec². Ans. 17.36
Suppose that the projectile is fired at a speed of 1024 ft/sec at an angle of 30° from the 2𝑥 4
83. Find the horizontal asymptote of the curve 𝑦 = 4 2 .
horizontal from a height of 2304 feet. 𝑥 −3𝑥 −1
Ans. y = 2
(a) Find parametric equations for the projectile.
84. A semi-ellipse and a parabola rests on the same base 60 meters wide and 20 m. high. Using
(b) Find the range r of the projectile—that is, the horizontal distance it travels before hitting
the common base as x-axis. Determine the difference of ordinates at a point 25 m. from the
the ground.
center of the base.
(c) Find the point and time at which the projectile reaches its maximum altitude.
Ans. 4.95
Ans. (a) x = 512√3 t, y = -16t² + 512t + 2304; t ≥ 0.
85. A plane has an equation of 4x + y + 8z + 33 = 0. Find the distance from point (1, 5, -3) to the
(b) 31,925 ft
plane.
(c) x = 15,189 ft, y = 6400 ft when t = 16
Ans. 2
56. If (r, θ) = (4, 7π/6) are polar coordinates of a point P, find the rectangular coordinates of P.
86. A parabola has an equation of x² = 6y + 10. Find the equation of the diameter of the parabola,
Ans. (−2√3, −2). which bisects chords having a slope of 4/3.
57. If (x, y) = (-1, √3) are rectangular coordinates of a point P, find three different pairs of polar Ans. x = 4
coordinates (r, θ) for P. 87. Find the equation of the plane through (2, 1, -3) parallel to the plane 3x + 4y + z = 4.
Ans. (2, 2π/3), (2, 8π/3) and (-2, -π/3) Ans. 3x + 4y + z = 7
58. Find a polar equation for the hyperbola x² - y² = 16. 88. A scalene triangle ABC is a triangle having three unequal sides. The distance from the
Ans. r² = 16 sec 2θ centroid of a triangle to the circumcenter is 6 units. How far is the centroid from the
59. If the length of the latus rectum of an ellipse is three-fourth of the length of the minor axis, orthocenter?
determine its eccentricity. Ans. 12
Ans. 0.661 89. Find the perpendicular distance between the planes 4x + y + 8z + 33 = 0 and 4x + y + 8z –
60. Determine the equation of the curve such that the sum of the distances of any point of the 30 = 0.
curve from two points whose coordinates are (-3, 0) and (3, 0) is always equal to 8. Ans. 7
Ans. 7x² + 16y² -112 = 0 90. Find the equation of the plane, which makes equal angles with the coordinate axes and
61. How far apart are the directrices of the curve 9x² + 25y² – 18x + 100y – 116 = 0 which cut a volume of 288 [Link] from the first octant.
Ans. 12.5 Ans. x + y + z = 12
62. Find the perimeter of an ellipse whose second eccentricity is 0.75 and distance between foci 91. A package of floor tiles contains 24 square meter tiles. Find how many packages should be
is 6 units. bought to cover a square ballroom floor whose side measures 64 meters.
Ans. 28.45 Ans. 171
3
63. The polar curve 𝑟 = is: 92. A gallon of latex paint can cover 500 square feet. Find how many gallon containers of paint
3+2 cos 𝜃
Ans. an ellipse should be bought to paint two coats on each wall of a rectangular room whose dimensions
64. Transform 𝑟 =
3
into cartesian coordinates are 14 feet by 16 feet (assume 8-foot ceilings).
3+2 cos 𝜃 Ans. 2 gallons
Ans. 5x² + 9y² + 12x - 9 = 0 93. A rectangular box is to be constructed to hold a new camcorder. The box is to have
65. If the curve Ax² + By² + F = 0 passes through (0, 3) and (4, 6), the curve is: dimensions 5 inches by 4 inches by 9 inches. Find the surface area of the box.
Ans. a hyperbola Ans. 202 square inches
66. Find the volume of the tetrahedron bounded by the coordinate planes and the plane 8x + 94. The volume V(x) of a box in terms of its height x is given by the function V(x) = x3 + 2x² – 8x.
12y + 4z – 24 = 0. Factor this expression for V(x).
Ans. 6 Ans. x(x+4)(x-2)
67. The equation of the plane (in rectangular coordinates) passing through the three points (1, 95. A manufacturer of metal washers needs to determine the cross-sectional area of each
3, 5), (2, 4, 4), and (3, 4, 5) is: washer. If the outer radius of the washer is R and the radius of the hole is r, express the
Ans. 2x – y + z – 4 = 0 area of the washer as a polynomial. Factor this polynomial completely.
68. Find the volume of a spherical wedge having a radius of 25 m. and a central angle of 71.62 Ans. π(R + r)(R + r) sq units
degree. 96. Marie has a rectangular board 12 inches by 16 inches around which she wants to put a
Ans. 6.7 cm3 uniform border of shells. If she has enough shells for a border whose area is 128 square
69. The graph, y = b – (x + A)2 passes through the points (0, - 3), (1, 0) and (c, 0). Find the inches, determine the width of the border.
value of c. Ans. 2
Ans. 3 97. The floor of a shed has an area of 90 square feet. The floor is in the shape of a rectangle
70. The line kx + (3 – k)y = 3(1 + k) passes through a fixed point P for any value of k. Find the whose length is 3 feet less than twice the width. Find the length and the width of the floor of
coordinates of P. the shed.
Ans. (4, 1) Ans. L = 12 ft, W = 7.5 ft
71. Where is the focus of the curve x² = - 12y? 98. A 50-foot supporting wire is to be attached to a 75-foot antenna. Because of surrounding
Ans. (0, 3) buildings, sidewalks, and roadways, the wire must be anchored exactly 20 feet from the
72. The equation of an asymptote of a hyperbola is equal to y = 2x which passes thru (5/2, 3). base of the antenna. How high from the base of the antenna is the wire attached?
Determine the length of the latus rectum. Ans. 45.8 feet
Ans. 16 99. The radius of the Moon is 1080 miles. Use the formula for the radius r of a sphere given its
73. Determine the length of the chord common to the circle x² + y² = 64 and x² + y² – 16x = 0. surface area A, r = √A / 4π to find the surface area of the Moon. Round to the nearest square
Ans. 13.86 mile.
Ans. 14,657,415 square miles
74. The equation of a curve is x² + y² = 25. Find the length of the sub-tangent at (-3, 4). 100. A holding pen for cattle must be square and have a diagonal length of 100 meters. Find the
Ans. 5.33 length of a side of the pen.
75. A curve has an equation of 9x² + 25y² = 225. Compute the second eccentricity of the curve. Ans. 50√2 meters
Ans. 1.33
GEOMETRY PRACTICE PROBLEMS
101. A board of length (3x4 + 6x² – 18) meters is to be cut into three pieces of the same length. 122. Complete each of the following statements:
Find the length of each piece. (a) If corresponding sides of two similar polygons are in the ratio of 4:3, then the ratio of their
Ans. x4 + 2x² – 6 meters perimeters is __?__
102. If the area of the rectangle is (15x² – 29x – 14) square inches, and its length is (5x² + 2) (b) The perimeters of two similar quadrilaterals are 30 and 24. If a side of the smaller
inches, find its width. quadrilateral is 8, the corresponding side of the larger is __?__
Ans. 3x – 7 inches (c) If each side of a pentagon is tripled and the angles remain the same, then each diagonal
103. The lateral surface of a cylinder varies jointly as its radius and height. Express this surface is __?__
area S in terms of radius r and height h. Ans. (a) 4/3 (b) 10 (c) tripled
Ans. S = 2πrh 123. In a right triangle, the hypotenuse has length 20 and the ratio of the two arms is 3:4. Find
104. Bailey’s rectangular dog pen for his Irish setter must have an end area of 400 square feet. each arm.
Also, the length must be 10 feet longer than the width. Find the dimension of the pen. Ans. 12 and 16
Ans. W=-5 + 5√17 ft., L=5 + 5√17 ft 124. Find the length of the altitude to the base of an isosceles triangle if the base is 8 and the
105. During the development stage of a new rectangular keypad for a security system, it was equal sides are 12.
decided that the area of the rectangle should be 285 square centimeters and the perimeter Ans. 8√2
should be 68 centimeters. Find the dimensions of the keypad. 125. In a rhombus, find (a) the length of a side s if the diagonals are 30 and 40; (b) the length of
Ans. 19 cm by 15 cm a diagonal d if a side is 26 and the other diagonal is 20.
106. The perimeter of a quadrilateral is 29 inches. The longest side is twice as long as the shortest Ans. (a) 25 (b) 48
side. The other two sides are equally long and are 2 inches longer than the shortest side. 126. (a) Find the distance d from the center of a circle of radius 17 to a chord whose length is 30.
Find the length of all four sides. (b) Find the length of a common external tangent to two externally tangent circles with radii
Ans. The side are 5 in, 7 in, 7 in, and 10 in. 4 and 9.
107. The orbit of the comet Kohoutek is about 44 astronomical units wide by 3600 astronomical Ans. (a) 8 (d) 12
units long. (One atronomical unit AU is the semimajor axis of the Earth’s orbit, about 127. Each leg of an isosceles trapezoid has length 18. If the base angles are 60° and the upper
92,600,000 miles.) Find the eccentricity of the orbit. base is 10, find the lengths of the altitude and the lower base
Ans. 0.9999 Ans. 9√3 and 28
108. Edmund Halley used Newton’s theory to calculate the orbit if the great comet of 1682. Ever 128. (a) Find the length of the leg of an isosceles right triangle whose hypotenuse has length 28.
since the comet’s return in 1758, it has been known as Halley’s comet. Last seen rounding (b) An isosceles trapezoid has base angles measuring 45°. If the upper base has length 12
the sun during the winter and spring of 1985-86, the comet is due to return again in the year and the altitude has length 3, find the lengths of the lower base and each leg
2062. Records of its passing go back 30 orbit cycles to 240 BC. A recent study indicated Ans. (a) 14√2 (b) 18 and 3, respectively
that a comet has made about 2000 cycles so far with about the same number to go before 129. (a) Find the area of a rectangle if the base has length 15 and the perimeter is 50.
the sun erodes it way completely. Compute the eccentricity of Halley’s comet if its orbit has (b) Find the area of a rectangle if the altitude has length 10 and the diagonal has length 26.
an ellipse 36.18 AU long by 9.12 AU wide. (c) Find the lengths of the base and altitude of a rectangle if its area is 70 and its perimeter
Ans. 0.97 is 34.
109. The distance between P1(2, 1, 5) and P2(-2, 3, 0) is Ans. (a) 150 (b) 240 (c) 7 and 10
Ans. 3√5 130. (a) Find the area of a square whose perimeter is 30.
110. Find the radius of the sphere x² + y² + z2 + 2x – 4y = 0 (b) Find the area of a square if the radius of the circumscribed circle is 10.
Ans. √5 (c) Find the side and the perimeter of a square whose area is 20.
Schaum’s Outline in Geometry (d) Find the number of square inches in a square foot.
Ans. (a) 56 ¼ (b) 200 (c) 8√5 (d) 144 in²
111. Find two angles such that: (a) The angles are supplementary and the larger is twice the
131. In a parallelogram, find the length of the altitude if the area is 54 and the ratio of the altitude
smaller. (b) The angles are complementary and the larger is 20° more than the smaller. (c)
to the base is 2:3.
The angles are adjacent and form an angle of 120°. The larger is 20° less than three times Ans. 6
the smaller. (d) The angles are vertical and complementary.
132. Find the area of (a) a rhombus in which the shorter diagonal has length 12 and an angle
Ans. (a) 60° and 120° (b) 35° and 55° (c) 35° and 85° (d) 45° each
measures 60°; (b) a regular hexagon with a side of length 6.
112. For each of the following, be represented by a and b. Obtain two equations for each case, Ans. (a) 72√3 (b) 54√3
and then find the angles. (a) The angles are adjacent, forming an angle of 88°. One is 36° 133. (a) Find the area of an isosceles trapezoid if the bases have lengths 22 and 10, and the legs
more than the other. (b) The angles are complementary. One is twice as large as the other.
have length 10. (b) Find the bases of an isosceles trapezoid if the area is 52√3 the altitude
(c) The angles are supplementary. One is 60° less than twice the other. (d) The angles are
has length 4√3, and each leg has length 8.
supplementary. The difference of the angles is 24°. Ans.(a) 128 (b) 17 and 9
Ans. (a) 62° and 26° (b) 60° and 30° (c) 100° and 80° (d) 78° and 102°
134. (a) Find the area of a rhombus if one diagonal has length 30 and a side has length 17.
113. Find the measure of each angle (a) Of a triangle if its angle measures are in the ratio of
(b) Find the length of a diagonal of a rhombus if the other diagonal has length 8 and the area
3:4:5. (b) Of a quadrilateral if its angle measures are in the ratio of 3:4:5:6. (c) Of a right of the rhombus is 52.
triangle if the ratio of the measures of its acute angles is 2:3 . Ans. (a) 240 (b) 13
Ans. (a) 45°, 60°, 75°(b) 60°, 80°, 100°, 120° (c) 36°, 54°, 90°
135. (a) The areas of two similar polygons are 80 and 5. If a side of the smaller polygon has
114. (a) Find the sum of the measures of the interior angles of a polygon of 9 sides (express your
length 2, find the length of the corresponding side of the larger polygon.
answer in straight angles and in degrees). (b) The corresponding diagonals of two similar polygons have lengths 4 and 5. If the area of
(b) Find the number of sides a polygon has if the sum of the measures of the interior angles the larger polygon is 75, find the area of the smaller polygon.
is 3600°.
Ans. (a) 8 (b) 48
(c) Is it possible to have a polygon the sum of whose angle measures is 1890°? 136. (a) Find the length of a side s of a regular pentagon if the perimeter p is 35.
Ans. (a) 1260°, (b) 22, (c) A polygon cannot have 12½ sides. So not a polygon (b) Find the length of the apothem a of a regular pentagon if the radius of the inscribed circle
115. (a) Find each exterior angle measure of a regular polygon having 9 sides. (b) Find each
is 21.
interior angle measure of a regular polygon having 9 sides. (c) Find the number of sides a
(c) In a regular polygon of five sides, find the measures of the central angle c, the exterior
regular polygon has if each exterior angle measure is 5°. (d) Find the number of sides a angle e, and the interior angle i.
regular polygon has if each interior angle measure is 165°. (d) If an interior angle of a regular polygon measures 108°, find the measures of the exterior
Ans. (a) 40°; (b) 140°; (c) 72 sides and (d) 24 sides
angle and the central angle and the number of sides.
116. Find each interior angle measure of a quadrilateral (a) if its interior angles are represented Ans. (a) 7 (b) 21 (c) 108° (d) 5
by x + 10,2x + 20, 3x – 50, and 2x – 20; (b) if its exterior angles are in the ratio 2:3:4:6. 137. In a regular hexagon, (a) find the lengths of the side and apothem if the radius is 12; (b) find
Ans. (a) 60°, 120°, 100°, 80°(b) 132°, 108°, 84°, 36°
the radius and length of the apothem if the side has length 8.
117. If two angles are in the ratio of 3:2, find the angles if (a) they are adjacent and form an angle
Ans. (a) 6√3 (b) 4√3
measuring 40°; (b) they are acute angles of a right triangle; (c) they are two angles of a 138. In a square, (a) find the lengths of the side and apothem if the radius is 16; (b) find the radius
triangle whose third angle measures 70°. and the length of the apothem if a side has length 10.
Ans. (a) 24° and 16° (b) 54° and 36° (c) 66° and 44°
Ans. (a) 8√2 (b) 5√2
118. Three angles are in the ratio of 4 :3: 2. Find the angles if (a) the first and the third are 139. In an equilateral triangle, (a) find the lengths of the radius, apothem, and side if the altitude
supplementary; (b) the angles are the three angles of a triangle. has length 6; (b) find the lengths of the side, apothem, and altitude if the radius is 9.
Ans. (a) 120°, 90°, and 60° (b) 80°, 60° and 40°
Ans. (a) 4√3 (b) 13½
119. Find the fourth proportional to (a) 2, 4, 6; (b) 4, 2, 6; (c) 3, 4; (d) b, d, c.
140. (a) Find the area of a regular hexagon if the length of the apothem is 5√3. (b) Find the area
Ans. (a) 12 (b) 3 (c) 24 (d) x = cd/b of a regular pentagon to the nearest integer if the length of the apothem is 20.
120. Find the positive mean proportional x between (a) 5 and 20; (b) ½ and 8/9. Ans. (a) 150√3 (b) 1453
Ans. (a) 10 (b) 2/3
141. (a) If the circumferences of two circles are in the ratio 2:3, find the ratio of the diameters and
121. (a) In two similar triangles, corresponding sides are in the ratio 3:2. Find the ratio of
the ratio of the areas.
corresponding medians. (b) The sides of a triangle are 4, 6, and 7. If the perimeter of a (b) If the areas of two circles are in the ratio 1:25, find the ratio of the diameters and the ratio
similar triangle is 51, find its longest side.
of the circumferences.
Ans. (a) 3/2, (b) 21
Ans. (a) 4/9 (b) 1/5
GEOMETRY PRACTICE PROBLEMS
142. (a) In two regular polygons having the same number of sides, find the ratio of the lengths of 161. What is the longest 6’ wide shuffle board court which will fit in a 20’ x 30’ rectangular room?
the apothems if the perimeters are in the ratio 5:3. Ans. 30⅞ feet
(b) In two regular polygons having the same number of sides, find the length of a side of the 162. A man leaves from the point where the prime meridian crosses the equator and moves forty-
smaller if the lengths of the apothems are 20 and 50 and a side of the larger has length 32.5. five degrees northeast by geographic compass which always points toward the north
(c) In two regular polygons having the same number of sides, find the ratio of the areas if geographic pole. He constantly corrects his route. Assuming that he walks with equal facility
the lengths of the sides are in the ratio 1:5. on land and sea, where does he end up and how far will he have travelled when he gets
(d) In two regular polygons having the same number of sides, find the area of the smaller if there?
the sides have lengths 4 and 12 and the area of the larger is 10,260. Ans. He arrives at the North Pole having traversed a distance of √2 × 10⁷ meters
Ans. (a) 5:3 (b) 13 (d) 1/25 (c) 1140 163. Near the town of Lunch, Nebraska there is a large triangular plot of land bounded by three
143. In a circle, (a) find the circumference and area if the radius is 6; (b) find the radius and area straight roads which are 855, 870, and 975 yards long respectively. The owner of the land,
if the circumference is 18π; (c) find the radius and circumference if the area is 144π. (Answer a friend of mine, told me that he had decided to sell half the plot to a neighbor, but that the
both in terms of p and to the nearest integer.) buyer had stipulated that the seller of the land should erect the fence which was to be a
Ans. (a) 113 (b) 254 (c) 75 straight one. The cost of fences being high, my friend naturally wanted the fence as short as
144. Find the circumference and area of the circumscribed circle and inscribed circle (a) of a possible. What is the minimum length the fence can be?
regular hexagon whose side has length 8; (b) of an equilateral triangle whose altitude has Ans. Exactly 600 yards
length 9√3. 164. Three hares are standing in a triangular field which is exactly 100 yards on each side. One
Ans. (a) 48π (b) 27π hare stands at each corner; and simultaneously all three set off running. Each hare runs
145. (a) Find the length of a 36° arc in a circle whose circumference is 45π. after the hare in the adjacent corner on his left, thus following a curved course which
(b) Find the radius of a circle if a 40° arc has a length of 4π. terminates in the middle of the field, all three hares arriving there are together. The hares
Ans. (a) 9π/2 (b) 18 obviously ran at the same speed, but just how far did they run?
146. (a) Find the area K of a 300° sector of a circle whose radius is 12. (b) Find the measure of Ans. Exactly 100 yards
the central angle of a sector whose area is 6π if the area of the circle is 9π. (c) Find the 165. A scalene triangle ABC which is not a right triangle has sides which are integers. If sin A =
radius of a circle if an arc of length 2π has a sector of area 10π 5/13’ find the smallest values for its sides, i.e., those values which make the perimeter a
Ans. (a) 120π (b) 240° (c) 10 minimum.
147. (a) Find the area of a segment if its central angle measures 60° and the radius of the circle Ans. a = 25, b = 16, c = 39
is 12. (b) Find the area of a segment if its central angle measures 90° and the radius of the 166. A one-acre field in the shape of a right triangle has a post at the midpoint of each side. A
circle is 8.c) Find each segment formed by an inscribed equilateral triangle if the radius of sheep is tethered to each of the side posts and a goat to the post on the hypotenuse. The
the circle is 8. ropes are just long enough to let each animal reach the two adjacent vertices. What is the
Ans. (a) 36√3 (b) 16π – 32 (c) ⅓(64π – 48√3) total area the two sheep have to themselves, i.e., the area the goat cannot reach?
148. Find the area of each segment formed by an inscribed regular polygon of 12 sides Ans. The two sheep have exactly one acre to themselves.
(dodecagon) if the radius of the circle is 12. 167. A divided highway goes under a number of bridges, the arch over each lane being the form
Ans. (a) 12π - 36 of a semi-ellipse with the height equal to the width. A truck is 6 ft. wide and 12 ft. high. What
149. If M is the midpoint of 𝑃𝑄 ̅̅̅̅, find the coordinates of (a) M if the coordinates of P and Q are is the lowest bridge under which it can pass?
P(3, 4) and Q(5, 8); (b) Q if the coordinates of P and M are P(1, 5) and M(3, 4). Ans. 13 ft 5 in high
Ans. (a) (4, 6) (b) (5, 3) 168. A cowboy is five miles south of a stream which flows due east. He is also 8 miles west and
150. Find the area of the triangle whose vertices are A(1, 2), B(7, 2), and C(5, 4). 6 miles north of his cabin. He wishes to water his horse at the stream and return home. What
Ans. 6 is the shortest distance he can travel and accomplish this?
151. Find the slope of the line through (-2, -1) and (4, 3). Ans. 8√5 or 17.9 miles
Ans. 2/3 169. An Origami expert started making a Nani-des-ka by folding the top left corner of a sheet of
152. Find the slope of the line whose equation is 3y - 4x = 15. paper until it touched the right edge and the crease passed through the bottom left corner.
Ans. 4/3 He then did the same with the lower right corner, thus making two slanting parallel lines. The
paper was 25 inches long and the distance between the parallel lines was exactly 7/40 of
Litton’s Problematical Recreations the width. How wide was the sheet of paper?
153. A forgetful physicist forgot his watch one day and asked an E.E. on the staff what time it Ans. 24 inches wide
was. The E.E. looked at his watch and said: “The hour, minute, and sweep second hands 170. The Ben Azouli are camped at an oasis 45 miles west of Taqaba. They decide to dynamite
are as close to trisecting the face they ever come. This happens only twice every 13 hours, the Trans-Hadramaut railroad joining Taqaba to Maqaba, 60 miles north of the oasis. If the
but since you probably haven’t forgotten whether you ate lunch, you should be able to Azouli can cover 18 miles a day, how long will it take them to reach the railroad?
calculate the time.” What time was it to the nearest second? Ans. d = 36 miles, requiring a two day trip.
Ans. 2 hours, 54 minutes, 35 seconds 171. A cross section through the center of a football is a circle x inches in circumference. The
154. The faces of a solid figure are all triangles. The figure has nine vertices. At each of six of football is x – 8 inches long from tip to tip and each seam is an arc of a circle ¾ x inches in
these vertices, four faces meet, and at each of the other three vertices, six faces meet. How diameter. Find x.
many faces does the figure have? Ans. 20.69 inches
Ans. 21 edges 172. A yang, ying, and yung is constructed by dividing a diameter of a circle, AB, into three parts
155. A new kind of atom smasher is to be composed of two tangents and a circular arc which is by points C and D, then describing on one side of AB semicircles having AC and AD as
concave towards the point of intersection of the two tangents. Each tangent and the arc of diameters and on the other side of AB semicircles having BD and BC as diameters. Which
the circle is 1 mile long. What is the radius of the circle? is larger, the central portion or one of the outside pieces?
Ans. 1437.45 feet Ans. All three are the same size, each being equal to ⅓ π R².
156. A spider and a fly are located at opposite vertices of a room of dimensions 1, 2, and 3 units. 173. A diaper is in the shape of a triangle with sides 24, 20, and 20 inches. The long side is
Assuming that the fly is too terrified to move, find the minimum distance the spider must wrapped around the baby’s wait and overlapped two inches. The third point is bought up to
crawl to reach the fly. the center of the overlap and pinned in place. The pin is to go through three thicknesses of
Ans. √18 units material. What is the area in which the pin may be placed?
157. In a room 40 feet long, 20 feet wide, and 20 feet high, a bug sits on an end wall at a point Ans. 2½ square inches
one foot from the other floor, midway between the sidewalls. He decides to go on a journey 174. An icicle forming from a dripping gutter is in the shape of a cone five times as long as it is
to a point on the other end wall which is one foot from the ceiling midway between the wide (at the top). A few hours later it has doubled in length and the generating angle has
sidewalls. Having no wings, the bug must make his trip by sticking to the surfaces of the also doubled. How does its present weight compare with previous weight?
room. What is the shortest route that the bug can take? Ans. The new icicle weighs almost 33 times as much as it weighed before.
Ans. 58 feet 175. A student beginning the study of trigonometry came across an expression of the form sin (X
158. A circle of radius 1 inch is inscribed in an equilateral triangle. A smaller circle is inscribed at + Y). He evaluated this as sin X + sin Y. Surprisingly he was correct. The values of X and Y
each vertex, tangent to the circle and two sides of the triangle. The process is continued with differed by 10°; what were these values, assuming that 0° < X < Y < 360°?
progressively smaller circles. What is the sum of the circumference of all circles. Ans. 175° and 185°
Ans. 5π inches 176. If the equal sides of an isosceles triangle are given, what length of the third side will provide
159. A farmer owned a square field measuring exactly 2261 yards on each side. 1898 yards from maximum area? (No calculus, please.)
one corner and 1009 yards from an adjacent corner stood a beech tree. A neighbor offered Ans. √2 times the length of one of the equal sides
to purchase a triangular portion of the field stipulating that a fence should be erected in a 177. One side of a triangle is 10 feet longer than another and the angle between them is 60°.
straight line from one side of the field to an adjacent side so that the beech tree was part of Two circles are drawn with these sides as diameters. One of the points of intersection of the
the fence. The farmer accepted the offer but made sure that the triangular portion was of two circles is the common vertex. How far from the third side is the other point of
minimum area. What was the area of the field the neighbor received, and how long was the intersection?
fence? Ans. At zero distance
Ans. 939,120 square yards and 2018 yards 178. There is one flag at the entrance to a racetrack and another inside the track, half a mile from
160. A coffee pot with a circular bottom tapers uniformly to a circular top with radius half that of the first. A jockey notes that no matter where is on the track, one flag is 3 times as far away
the base. A mark halfway up the side says “2 cups.” Where should the “3 cups” mark go? as the other. How long is the track?
Ans. 3 1/37 cups Ans. 1908π ft or about 3.36 furlongs
GEOMETRY PRACTICE PROBLEMS
PROBABILITY AND STATISTICS
1. There are three flights from Houston to Chicago, four flights from Chicago to 17. The table below displays the results of a survey regarding the number of pets
Memphis and five flights from Memphis to Atlanta. How many choices of flights each student in a class has. The average number of pets per student in this
include the Houston-Chicago-Memphis-Atlanta connection? class is 2. Find the value of K.
Ans. 12 No. of Pets 0 1 2 3 4 5
2. A calibration study needs to be conducted to see if 20 scales are giving the No. of Students 4 6 10 0 K 2
same weights. How many ways may be selected to perform a preliminary Ans. 4
study? 18. From the given data shown:
Ans. 15,504 Score 1 2 3 4 5
3. Industrial engineers are timing to workers to rank their speed at assembling Frequency 14 15 14 17 10
computers. There are 10 workers. How many different permutations are Determine the standard deviation.
possible? Ans. 1.349
Ans. 3628800 19. The mean of the numbers x1, x₂, x3 . . . x8 is 9. If the mean of x₂, x3, x6, x7, x8 is
4. Suppose a lot consisting of 100 items contains 5 defectives. If one item is 10, find the mean of x1, x4, and x5.
randomly selected, the probability P(item is defective) is: Ans. 7.33
Ans. 0.05 20. An experiment consists of selecting three items from a box of several items. The
5. Suppose 3 items are inspected and if at least one defective is found, the lot will three items are classified defective (D) or non-defective (N) as they are
be 100% inspected. Otherwise, the lot will be passed on. How likely it is that a selected. Give the sample space for this experiment. Which outcomes are in the
lot containing 5 defectives will be passed on? events A, B, C? (a) A, Exactly one item defective? (b) B, at least two defectives
Ans. 0.855999 in the three. (c) C, at most one defective in the three.
6. The probability that a construction generator will operate satisfactorily for 5 yrs Ans. (a) A = {DNN, NDN, NND} (b) B= {NDD, DND, DDN, DDD} (c) C = {NNN,
is 0.80 and that a welding machine will operate satisfactorily over the same DNN, NDN, NND}
period of time is 0.75. Find the probabilities that in a 5-year period both 21. An experiment consists of inspecting items from a production line until a
generator and welding machine operate satisfactorily. defective (D) is found. Give the sample space for this experiment. Which
Ans. 0.60 outcomes are in the events A, B, and C? (a) A, Exactly one item is inspected.
7. It is known that 60% of a class is female, 50% is majoring in chemical or (b) B, at least two are inspected. (c) C, At most five items are inspected.
electrical engineering, and 25% is female and majoring in chemical or electrical Ans. (a) A = {D} (b) B= {ND, NND, NNND, NNNND, …} (c) C ={D, ND, NND,
engineering. What percentage of class is female or majoring in chemical or NNND, NNNND}
electrical engineering? 22. In how many different ways can one make a first, second, third, and fourth
Ans. 85% choice among 10 firms leasing construction equipment?
8. Suppose 1% of parts have type 1 defect and 2% have 2 defect and 0.5% have Ans. 5040
both types of defects. A part is known to have type 1 defect. What is the 23. An electronic controlling device requires six identical memory chips. In how
probability that it has type 2 defect? many ways can this mechanism be assembled using six given chips?
Ans. 0.25 Ans. 720
9. A box contains 5 defective and 195 non‐defective cell phones. A quality control 24. In how many different ways can the director of a research laboratory choose two
engineer selects two cell phones at random with replacement. (What is the chemical engineers from among five and two industrial engineers from among
probability that (a) neither is defective? (b) exactly 1 is defective? (c) both are four applicants?
defective? Ans. 60
Ans. (a) 0.950625 (b) 0.04875 (c) 0.000625 25. It is common in many industrial areas to fill boxes full of product. This occurs in
10. If a die is rolled 5 times, what is the probability of obtaining five 6s in a row? an industry where the product is used in the home; for example, detergent.
Ans. (1/6)⁵ These machines fill to specification (A), under‐fill (B), or over‐fill (C). The
11. A dart target board consists of a center circle having a radius of 2 inches a practice of under‐filling is the one that we try to avoid. Suppose that it is known
square section with dimension of 6 in x 6 in and the radius of the biggest circle that P(B) = 0.001 and P(A) = 0.990. (a) Find P(C). (b) What is the probability
is 6 in. The three cross-sections have the same center at 0. Assuming the dart that the machine does not under-fill? (c) What is the probability that the machine
is equally likely to hit any point inside the target. Find the probability that a dart over-fill s or under-fills?
thrown at the circular target will hit the area outside the square. Ans. (a) 0.009 (b) 0.999 (c) 0.010
Ans. 0.682 26. Specifications are given for the weight of a packaged product. The package is
12. An employee of a large company, Toyota Motors Inc., is promoted to rejected if it is too heavy or too light. Historical data suggests that 0.95 is the
management and will be transferred within 6 months. The employee is told that probability that the product meets weight specifications and 0.002 is the
there is a 33% probability of being transferred to Cebu and a 50% probability of probability that the product is too light. For each single packaged product, the
being transferred to Davao. What is the probability that the employee will be manufacturer invests $20 in production and the purchase price is $25.
transferred to Cebu or Davao? (a) What is the probability that a package chosen at random from the
Ans. 0.83 production line is too heavy?
13. The probability that both stages of a two-stage missile will function correctly is (b) For each 10,000 packages sold, what profit is received by the
0.95. The probability that the first stage will function correctly is 0.98. What is manufacturer if all packages meet weight specifications?
the probability that the second stage will function correctly given that the first (c) Assuming that all defective packages are rejected and rendered
one does? worthless, how much is the profit reduced on 10,000 packages due to failure to
Ans. 0.97 meet weight specifications?
14. A machine has a probability of producing a defective equal to 0.05 every time it Answers: (a) 0.048, (b) $50,000 (c) $12,500
produces a product. If three of its products are selected randomly and Ans. 0.048
independently during a shift, what is the probability that a quality engineer will 27. A component fails a particular test (A) 15% of the time or a component displays
find: (a) no defective in the three? (b) 1 defective in the three? (c) 2 defective in strain but does not fail the test (B) 25% of the time.
the three? (d) 3 defective in the three? (a) What is the probability that the component does not fail the test?
Ans. (a) 0.857375 (b) 0.135375 (c) 0.007125 (d) 0.000125 (b) What is the probability that a component works perfectly (neither displays
15. A game is played as follows. You pay $1 to play. A coin is flipped four times. If strain nor fails the test)?
four tails or four heads are obtained, you get your 1$ back plus $5 more. (c) What is the probability that the component either fails or shows strain in
Otherwise you forfeit your $1. What it the mathematical expectation? the test?
Ans. -0.25 Answers: (a) 0.85 (b) 0.6 (c) 0.4
16. An experiment has a continuous sample space equal to 15≤X≤20. If events A 28. A set of events are collectively exhaustive if at least one of the events occur
and B are A = {X: 15 ≤ X ≤ 17}, B = {X: 16 ≤ X ≤ 19}, find: (a) A ∩ B. (b) A ∪ B when the experiment is performed. (True or False) (A) A die is rolled and the
(c) Ac events A₁, the face 1 turns up, … , A₆, the face 6 turns up are collectively
Ans. (a) {X: 16 ≤ X ≤ 19}; (b) {X: 15 ≤ X ≤ 19} (c) {X: 17 < X ≤ 20} exhaustive events. (b) The events A, a face that is an even number turns up
and B, a face that is on odd number turns up are collectively exhaustive. (c) A₁
PROBABILITY AND STATISTICS
is the event that a 1 or 2 turns up, A₂ is the event that a 3 or 4 turns up, A₃ is 41. When ordering a computer, there are 3 choices of hard drive, 2 choices of the
the event that a 5 or 6 turns up. The events A₁, A₂ and A₃ are collectively amount of memory, 3 choices of video card, and 2 choices of monitor. How
exhaustive. many different computers could be ordered?
Ans. a, b, and c are all true Ans. 36
29. A quality engineer selects 3 iPods from a box that contains 20 iPods of which 3 42. Twelve engineers have applied for a management position at a large firm. Three
are defective. What is the probability that the first is defective and the others are of them will be selected as finalists for the position. In how many ways can this
not defective? selection be made?
Ans. 0.119 Ans. 220
30. Two thirds of people in a party are lawyers, whose average I.Q. is 120. The rest 43. A computer password consists of eight characters. A character is any lower-
are engineers whose I.Q. is 180. What is the average I.Q. of all persons in the case letter or digit. How many passwords are available?
room? Ans. 36^8 or 2.82111E+12
Ans. 140 44. As a project, a freshmen communication major is assigned the following project:
31. A set has 5 items and it has a range of 7. The set is composed of the following: Randomly select 125 students and count how many are talking on a cell phone
{1, 2, M, 5, M² } with M > 0 at a randomly chosen time. The major counts 83 students that are talking on a
Find the average of the number set. cell phone. What is the frequency probability that a student randomly selected at
Ans. 3.76 random time will be talking on a cell phone?
32. How many different signals each consisting of 6 flags hung in a vertical line can Ans. 0.664
be formed by 4 identical red flags and 2 identical blue flags? 45. A professor of international studies is asked to estimate the probability that a
Ans. 15 terrorist act will be performed in a large European city within the next year. The
33. You know that the extension of a private telephone number is 272 but you have professor states that there is a 10% chance. What is the subjective probability
forgotten the last 4 digits. You can only recall that the last 4 digits are 3, 6, 8, that this event occurs?
and 9 but you do not know the order. What is the maximum number of Ans. 0.10
telephone calls you will need to make in order to dial the correct number? 46. A box of bolts contains 4 thick bolts, 3 medium bolts and 3 thin bolts. A box of
Ans. 24 nuts contains 5 that fit thick bolts, 5 that fit medium bolts, and 5 that fit thin bolts.
34. A music school produced a number of musicians which includes 3 drummers, 4 One bolt and one nut are chosen at random. What is the classical probability
trumpet players and 5 pianists. How many different jazz trios can be formed that the nut fits the bolt?
from this batch of musicians if each trio consists of a drummer, a trumpet player Ans. 1/3
and a pianist? 47. The probabilities that a consumer testing service will rate a new antipollution
Ans. 60 device for cars very poor, poor, fair, good, very good, or excellent are 0.06,
35. A production line produces flash drives of which 1% are defective. A quality 0.14, 0.20, 0.3, 0.22, and 0.08. What are the probabilities that it will rate the
engineer selects a sample of three. What is the probability that the sample device (a) very poor, poor, or fair? (b) very good and excellent?
contains three defectives? Ans. (a) 0.40 (b) 0
Ans. 0.000001 48. The probabilities are 0.80, 0.70, and 0.65 that a family, randomly chosen as a
36. One thousand copper rods have the properties shown in the following table. part of a sample survey in a large city, owns a flat screen television or a
Diameter computer or both. What is the probability that a family in this area own one or
the other or both?
Length Too thin OK Too thick
Ans. 0.85
Too short 10 5 5
49. In a group of 150 graduate engineering students, 90 are enrolled in an
OK 40 902 7 advanced course in statistics, 60 are enrolled in a course in operations
Too long 4 20 7 research, and 40 are enrolled in both. How many of these students are not
(a) If the rod meets the length specifications, find the probability the rod meets enrolled in either course?
the diameter specifications. (b) If the rod meets the diameter specifications, find Ans. 40
the probability the rod meets the length specifications. (c) Find the unconditional 50. Students at University of the Philippines are classified as being freshmen,
probability that the rod meets the length specifications. sophomores, juniors, or seniors, and also according to whether they are male or
Ans (a) 0.95 (b) 0.97 (c) 0.949 female. Find the total number of possible classifications for the students of this
37. On the East Coast, it is known from health records that the probability of college.
selecting an adult over 40 years of age with cancer is 0.05. The probability of Ans. 8
diagnosing a person with cancer as having the disease is 0.78 and the 51. The standard configuration for a license plate is 3 digits followed by 3 letters.
probability of incorrectly diagnosing a person without cancer as having the How many different license plates are possible if the digits and letters can be
disease is 0.06. Find the probability that repeated?
(a) a person is diagnosed as having cancer. Ans. 17,576,000
(b) a person diagnosed as having cancer actually has the disease. 52. The LTO issues license plates consisting of letters and numbers. There are 26
(c) a person diagnosed as having a cancer does not have the disease. letters may be repeated. There are 10 digits and the digits may be repeated.
Ans. (a) 0.096 (b) 0.40625 (c) 0.59375 How many possible license plates can be issued with two letters followed by
38. A group of engineers decide to play the game of craps. A pair of dice is rolled in three numbers?
this game and the sum to appear on the dice is of interest. What is the Ans. 676,000
mathematical expectation of the sum to appear when the dice are rolled? 53. A process manufactures aluminium cans. The probability that a can has a flaw
Ans. 7 on the side is 0.02, the probability that it has a flaw on the top is 0.03, and the
39. The number of defective welds in a length of pipe 0 through 6 with the following probability that it has a flaw on the top and the side is 0.01. What is the
probabilities. probability that a can will have (a) flaw on its side, given that it has a flaw on the
X 0 1 2 3 4 5 6 top? (b) a flaw on the top given it has a flaw on the side?
P(X) 0.60 0.30 0.05 0.02 0.01 0.01 0.01 Ans. (a) 0.33 (b) 0.50
Find the expected value of the number of defective welds. 54. The following table gives the distribution of the number of defects X in a
Ans. 0.61 randomly chosen printed-circuit board.
40. The probability that a relay remains open is 0.5. An electrical circuit consist of X 0 1 2 3
three such relays. The number of relays that remain open in three are 0, 1, 2, or P(X) 0.6 0.2 0.15 0.05
3. The number of relays that remain open is represented by X. The number of Find the mathematical expectation of X.
relays that remain open has the following values with the probabilities. Ans. 0.65
X 0 1 2 3 55. Of the microcomputers manufactured by a certain process, 5% are defective.
P(X) 0.125 0.375 0.375 0.125 Four of the microcomputers are chosen at random. Assume they function
independently. What is the probability that they all work?
Find the expected value of the number of open relays.
Ans. 1.50 Ans. 0.81
PROBABILITY AND STATISTICS
56. A chemical supply company ships a chemical in 5-gallon drums. X represents 67. The manager of job shop does not know the probability distribution of the time
the number of drums ordered by a randomly chosen customer. X has the required to complete an order. From past performance she has been able to
following distribution. estimate the mean and standard deviation to be 14 days and 2 days. Find the
X 1 2 3 4 5 interval so that the probability is at least 75% that the order is finished in that
P(X) 0.1 0.1 0.2 0.2 0.4 time.
Find the mathematical expectation of X. Ans. 10 to 18 days
Ans. 3.7 68. The daily production of electric motors at a particular factory averages 120 with
57. A Ford has engines in three sizes. Of the Ford cars sold, 50% have the smallest a standard deviation of a 5. (a) What fraction of the days will have a production
engine, 40% have the medium engine. Of the cars with the smallest engine, between 110 and 130? (b) Find the shortest interval certain to contain at least
15% fail an emissions test within two years of purchase. The failure figure for 96% of the daily production of electric motors.
medium size engines is 10%, and the failure figure for the largest engines is Ans. (a) ¾ days (b) 95 to 145
5%. A record for a failed emission test is randomly chosen. (a) What is the 69. Chemical engineers claim that 70% of all Americans have perfluorooctanoic
probability that it is for a Ford with the smallest engine? (b) What is the acid (PFOA) in their bloodstream. This chemical is a likely human carcinogen.
probability that it is for a Ford with the smallest engine? Given that the 70% figure is correct, find the following blood tests for PFOA in
Ans. (a) 0.12 (b) 0.625 human blood. (Thirty Americans were randomly tested for PFOA.)
58. A resistor in a certain circuit is specified to have a resistance in the range of 99 (a) What is the mean or expected number that you would find in a sample of 30
ohms to 101 ohms. An engineer obtains two resistors. The probability that both who tested for PFOA?
of them meet the specification is 0.30, the probability that exactly one of them (b) What is the standard deviation of the number found in samples of 30 who
meet the specification is 0.60, and the probability that neither of them meets the test positive for PFOA?
specifications is 0.10. Let X represent the number of resistors that meet (c) What is the probability of finding 15 or fewer in a sample of 30 who test
specifications. Find the mathematical expectation of X. positive for PFOA?
Ans. 1.2 (d) What is the probability of finding 25 or more in a 30 who test positive for
59. A quality control engineer has decided to inspect play stations until 4 defectives PFOA?
are found. Suppose 1% of the play stations are defective. Let X stand for the Ans. (a) 21 (b) 2.5 (c) 0.0169 (d) 0.0766
number of inspections to find 4 defectives. (a) What is the mean number of 70. An industrial firm supplies 10 manufacturing plants with a certain chemical. The
inspections needed to find 4 defectives? (b) What is the standard deviation of probability that any one firm calls in an order on a given day is 0.3, and this is
X? the same for all 10 plants. Let X represent the number of plants to call in an
Ans. (a) 400, (b) 199 order on a given day. (a) Find the mean of X, (b) the standard deviation of X, (c)
60. Compute the standard deviation of the normal distribution that approximates a the probability that the number of plants calling in orders is at most 3, (d) and
binomial distribution. There are 60 trials with a probability of failure of 0.25. the probability that the number of plants calling in orders is at least 3.
Ans. 3.35 Ans. (a) 3, (b) 1.45 (c) 0.650 (d) 0.617
61. A lathe machine in a mechanical shop breaks down an average of 4 times per 71. The mean score of Mathematics in a certain exam was 60, and the standard
year. Using Poisson’s distribution, find the probability that at most 1 breakdown deviation was 15. If your score is within 2 standard deviations of the mean, what
will occur each year. is the lowest score you would receive?
Ans. 0.1079 Ans. 30
62. A fair die is thrown 5 times. On any one throw, outcome 1 is that an even 72. A large set of data has a normal distribution with a mean of 56 and a standard
number appears, outcome 2 is that a 1 or 3 appears, and outcome 3 is that 5 deviation of 5. Determine the probability the intervals between 51 to 61.
appears. Find the probability that in the 5 throws, outcome 1 occurs twice, Ans. 68.2%
outcome 2 occurs twice, and outcome 3 occurs once. 73. The amount of time that a teenager plays videogames in any given week is
Ans. 0.138848 normally distributed. If a teenager plays videogames an average of 15 hours per
63. Based on past experience, the makeup of Dr. Stephens’ statistics for engineers week with a standard deviation of 3 hours, what is the probability of a teenager
class has consisted of 20% civil engineers, 30% chemical engineers, 40% playing videogames between 9 and 21 hours?
industrial engineers, and 10% electrical engineers. Given that these Ans. 95.4%
probabilities still hold, what is the probability that her 2010 classes will 74. At the La Salle College, there are 16 students in English Club, 16 students in
consistent of 25 civil engineers, 30 chemical engineers, 40 industrial engineers, Science Club, and 20 students in Math Club. Of these students, there are 5
and 10 electrical engineers? students in both English and Science Clubs, 6 students in both the Science and
Ans. 0.00074 Math Clubs, and 8 in both English and Math Clubs. If only 2 students are all in
64. A large engineering company employs a large number of individuals with the three clubs, how many students are in at least one of the clubs?
engineering backgrounds. Of these, 2% have Ph.D degrees, 12% have masters Ans. 20
degrees, 38% have bachelors degrees, and the remainder have technical 75. The average score for a Mathematics test is 77 and the standard deviation is 8.
background, but no degree. The educational backgrounds are coded in the Compute the probability that any one student scored between 61 and 93.
personnel department as 1, 2, 3, or 4 depending on whether the employee has Ans. 95.4%
a Ph.D, masters degree, bachelors degree, or no degree. Random variable X 76. About 68.2% of the scores fall within the normal curve, which ranges from 50 to
represents the coded educational background. Give the probability distribution 80 and is symmetric about the mean. What is the standard deviation of the
of X, the cumulative probability distribution of X, and find the P(X < 3). scores in this distribution?
Ans. 0.14 Ans. 15
65. Daily sales records for an engineering consulting firm shows that it will attract 0, 77. On a standard test, Peter received a score of 85, which was exactly 2 standard
1, or 2 consulting jobs per day with probabilities as given below. deviations above the mean. If the standard deviation of the test is 4, what is the
mean for this test?
Consulting jobs 0 1 2
Ans. 77
Probability 0.10 0.75 0.15
78. Samsung, a computer chip manufacturer, has found that only 1 out of 2000
Find the probability that it will attract 0, 1, 2, 3 or 4 consulting jobs in a two-day chips is defective. A certain company ordered a shipment of chips. How many
period, assuming the number of consulting jobs are independent from day to chips will the company ordered before the probability that at least one chip is
day. defective is 45%?
Ans. P(X = 0) = 0.01; P(X = 1) = 0.15; P(X = 2) = 0.5925; P(X = 3) = 0.225; P(X Ans. 1194
= 4) = 0.0225 79. Of 50 buildings in an industrial park, 12 have electrical code violations. If 10
66. Daily sales records for a computer manufacturing firm show that it will sell 0, 1, buildings are selected at random for inspection, (a) what is the expected
2, or 3 mainframe computer systems with the following probabilities. number to have electrical code violations? (b) What is the standard deviation of
x 0 1 2 3 the number to have electrical code violations? (c) What is the probability that 3
p(x) 0.6 0.15 0.15 0.10 or fewer have electrical code violations?
Find the expected value, variance, and standard deviation for daily sales. Ans. (a) 2.4 (b) 1.22 (c) 0.821
Ans. μ = 0.75, σ² = 1.0875, and σ = 1.0428
PROBABILITY AND STATISTICS
80. If the standard deviation of a set of observation is 0, you can conclude probabilities: (a) the probability of no accidents in a given week, (b) the
Ans. That all observations are the same value. probability of at most 4 accidents in a given week, and (c) the probability of at
B. That there is no relationship between the observations. least 2 accidents in a given week.
C. That the average value is 0. Ans. (a) 0.0497871 (b) 0.815263 (c) 0.801
D. That a mistake in arithmetic has been made. 90. The time required for the first stage on an assembly line is normally distributed
81. Which of the following are true statements? with a mean of 32 minutes and variance of 6 minutes, while the time required for
I. In an experiment some treatment is intentionally forced on one group to an independent second stage is normally distributed with a mean of 43 minutes
note the response. and variance of 10 minutes. What is the mean and the variance of the two
II. In an observational study information is gathered on an already existing stages?
situation. Ans. 75, 4
III. Sample surveys are observational studies, not experiments. 91. From the set of numbers 14, 8, 6, 2, find the mean absolute deviation.
Ans. I, II and III Ans. 3.5
82. Which of the following are true statements about sampling error? 92. A simple random sample is defined by
I. Sampling error can be eliminated only if a survey is both extremely well Ans. The method of selection
designed and extremely well conducted. B. Examination of the outcome
II. Sampling error concerns natural variation between samples, is always C. Whether or not a random number generator is used
present, and can be described using probability. D. How representative the sample is of the population
III. Sampling error is generally smaller when the sample size is larger. 93. The speeds of cars passing through an automated E-Z Pass toll booth have a
Ans. II and III nearly normal distribution with a mean of 5 mph and a standard deviation of 0.6
82. Three randomly chosen cars of a particular model are subjected to sideswipe mph. What is a speed at the 30th percentile? (z score at the 30th percentile =
(shallow-angle) collisions and the monetary damages are noted. Assuming that 0.524)
the monetary damages in this type of accident are normally distributed, a 95% t- Ans. 4.69 mph
based confidence interval (with t = 4.303) for the mean is found. Which of the 94. The probabilities that the light bulb of the projector used in Dr. Stephens’
following is the correct statement? statistical engineering software course will last less that 40 hours, 40 to 80
Ans. The confidence interval is valid. hours, or more that 80 hours are 0.3, 0.5, and 0.2. Find the probability that
B. The confidence interval is invalid because the sample size is too small. among 8 such bulbs, 2 will last less than 40 hours, 5 will last between 40 and 80
C. The confidence interval is invalid because np = 3 < 10. hours, and 1 will last more than 80 hours.
D. The confidence interval is invalid because t = 3.182 should have been Ans. 0.0945
used. 95. Items under inspection are subject to two types of defects. About 70% of the
83. Under the condition N/S < 0.05 where S is the number of successes and N is items in a large lot are defect-free, 20% have a type A defect, and 10% have a
the population size, the binomial distribution and the hypergeometric distribution type B defect. Six of the items are randomly selected. Find the probability that 3
give probabilities that are very close to one another. Suppose a shipment of 120 have no defects, 1 has a type A defect, and 2 have a type B defect.
burglar alarms contain 5 that are defective. If 3 of these alarms are selected and Ans. 0.041
shipped to a costumer, calculate the probability that 1 of the 3 will be defective 96. For a large shipment of integrated circuits the probability of failure for any one
using (a) the hypergeometric distribution and (b) the binomial distribution. chip is 0.01. Assuming the assumptions underlying the binomial distribution are
Ans. (a) 0.1167 (b) 0.1149 met, find the probability that at most two chips fail in a random sample size 20.
84. A recruiting firm finds that 20% of the applicants have advanced training in Ans. 0.999
computer programming. Applicants are interviewed sequentially and are 97. From the given data of a distribution of breakdown per week of a certain
selected at random from the pool of applicants. (a) Find the probability that the machine, determine the number of times this machine is expected to breakdown
first applicant having advanced training is found on the fifth interview. (b) Find per week over a period of time.
the mean number of applicants needed before finding the first with advanced Probability Breakdown/week
training in computer programming. 0.11 0
Ans. (a) 0.0819 (b) 5 0.18 1
85. A structural engineer performs a test of weld strength that involves loading 0.26 2
welded joints until a fracture occurs. For a certain type of weld, 95% of the 0.35 3
fractures occur in the weld itself while the other 5% occur in the beam. A 0.42 4
number of welds are tested. Let X be the number of tests up to and including Ans. 3.43
the first test that results in a beam fracture. (a) What is the distribution of X? (b)
What is the probability that X is less than 10? (c) What is the mean of X? Fifty challenging problems in probability
Ans. (a) P(X) = 0.95X – 1(0.05) (b) P(X < 10) =0.3698 (b) 20 98. A drawer contains red socks and black socks. When two socks are drawn at
86. A petroleum engineer knows that within a certain region, the probability that a random, the probability that both are red is ½ (a) How small can the number of
drilling will lead to discovering oil is 0.45. Let X represent the drilling on which oil socks in the drawer be? (b) How small if the number of black socks is even?
is discovered for the third time. The drillings are independent of one another. Ans: (a) 4 (b) 21
(a) Find P(X ≤10). 99. A three-man jury has two members each of whom independently has probability
(b) Find the mean and variance of the random variable X. p of making the correct decision and a third member who flips a coin for each
Ans. (a) P(X ≤10) = 0.90044 (b) μ = 6.67, σ² = 8.148, σ = 2.85 decision (majority rules) A one-man jury has probability p of making the correct
87. In a test of weld length, 80% of tests result in a fracture in the weld, while the decision. Which jury has the better probability of making the correct decision?
other 20% result in a fracture in the beam. Let X represent the number of tests Ans: The two juries have the same chance of a correct decision.
up to and including the third beam fracture. (a) What is the probability that X is 100. On the average, how many times must a die be thrown until one gets a 6?
equal to or less than 10? (b) Find the mean and variance of the random variable Ans: 6
X. 101. Chuck-a-Luck is a gambling game often played at carnivals and gambling
Ans. (a) P(X ≤10) = 0.3222 (b) μ = 15, σ² = 60, σ = 7.75 houses. A player may bet on any one of the numbers 1, 2, 3, 4, 5, 6. Three dice
88. An industrial engineer knows that 0.03% of plastic containers manufactured by are rolled. If the player's number appears on one, two, or three of the dice, he
a certain process have small holes that render them unfit for use. Let X receives respectively one, two, or three times his original stake plus his own
represent the number of containers in a random sample of 10,000 that have this money back; otherwise he loses his stake. What is the player's expected loss
defect. Find the following: (a) P(X = 3) (b) P(X ≤ 2) (c) P(1 ≤ X < 4) (d) mean per unit stake? (Actually the player may distribute stakes on several numbers,
and the standard variation of X. but each such stake can be regarded as a separate bet.)
Ans. (a) 0.224 (b) 0.4232 (c) 0.5974 (d) The mean = 3 and standard deviation = Ans: 8% per play
√3 102. Mr. Brown always bets a dollar on the number 13 at roulette against the advice
89. An industrial engineer knows that the number of industrial accidents averages 3 of Kind Friend To help cure Mr Brown of playing roulette, Kind Friend always
per week in the industry in which she is employed. X = the number of accidents bets Brown $20 at even money that Brown will be behind at the end of 36 plays.
per week, is known to have a Poisson distribution. Find the following How is the cure working? (Most American roulette wheels have 38 equally likely
PROBABILITY AND STATISTICS
numbers. If the player's number comes up, he is paid 35 times his stake and 115. A bread salesman sells on the average 20 cakes on a round of his route. What
gets his original stake back; otherwise he loses his stake) is the chance that he sells an even number of cakes? (We assume the sales
Ans: Kind Friend will be cured first. follow the Poisson distribution.)
103. We often read of someone who has been dealt 13 spades at bridge. With a Ans: 0.568
well-shuffled pack of cards, what is the chance that you are dealt a perfect hand 116. What is the least number of persons required if the probability exceeds ½ that
(13 of one suit)? (Bridge is played with an ordinary pack of 52 cards, 13 in each two or more of them have the same birthday? (Year of birth need not match.)
of 4 suits, and each of 4 players is dealt 13 ) Ans: 23
Ans: 4 × 13!39!/52! 117. You want to find someone whose birthday matches yours. What is the least
104. The game of craps, played with two dice, is one of America's fastest and most number of strangers whose birthdays you need to ask about to have a 50-50
popular gambling games. Calculating the odds associated with it is an chance?
instructive exercise. Ans: 253
The rules are these. Only totals for the two dice count. The player throws the 118. If r persons compare birthdays in the pairings problem, the probability is PR that
dice and wins at once if the total for the first throw is 7 or 11, loses at once if it is at least 2 have the same birthday. What should n be in the personal birthmate
2, 3, or 12. Any other throw is called his "point." If the first throw is a point, the problem to make your probability of success approximately PR?
player throws the dice repeatedly until he either wins by throwing his point again Ans: n = r(r – 1)/2
or loses by throwing 7. What is the player's chance to win? 119. How thick should a coin be to have a 1/3 chance of landing on edge?
Ans: 0.49293 but anticipate 0.27071 (The examiner could get wrong) Ans: 35% as thick as the diameter of the coin
105. Two strangers are separately asked to choose one of the positive whole 120. Shuffle an ordinary deck of 52 playing cards containing four aces. Then turn up
numbers and advised that if they both choose the same number, they both get a cards from the top until the first ace appears. On the average, how many cards
prize. If you were one of these people, what number would you choose? are required to produce the first ace?
Ans: 3 and 7 are popular choices Ans: 10.6th
106. Coupons in cereal boxes are numbered 1 to 5, and a set of one of each is 121. (a) A railroad numbers its locomotives in order, 1, 2, … , N. One day you see a
required for a prize. With one coupon per box, how many boxes on the average locomotive and its number is 60. Guess how many locomotives the company
are required to make a complete set? has.
Ans: 11.42 (b) You have looked at 5 locomotives and the largest number observed is 60.
107. When 100 coins are tossed, what is the probability that exactly 50 are heads? Again guess how many locomotives the company has.
Ans: 0.07959 or about 0.08 Ans: (a) 119 (b) 71
108. A, B, and C are to fight a three-cornered pistol duel. All know that A's chance of 122. If a stick is broken in two at random, what is the average length of the smaller
hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their piece?
choice of target in succession in the order A, B, C, cyclically (but a hit man loses Ans: ¼ of the length
further turns and is no longer shot at) until only one man is left unhit. What 123. A bar is broken at random in two places. Find the average size of the smallest,
should A's strategy be? of the middle-sized, and of the largest pieces.
Ans: A fires his first shot into the ground and then tries to hit B with his Ans: 1/9, 5/18, 11/18
next shot. 124. Under the conditions of the previous matching problem, what is the probability
109. Two urns contain red and black balls, all alike except for color Urn A has 2 reds of exactly r matches?
and 1 black, and Urn B has 101 reds and 100 blacks. An urn is chosen at Ans: 0.368 for large n.
random, and you win a prize if you correctly name the urn on the basis of the 125. What is the probability that the quadratic equation x² + 2bx + c = 0 has real
evidence of two balls drawn from it. After the first ball is drawn and its color roots?
reported, you can decide whether or not the ball shall be replaced before the Ans: 1
second drawing. How do you order the second drawing, and how do you decide 126. Starting from an origin 0, a particle has a 50-50 chance of moving 1 step north
on the urn? or 1 step south, and also a 50-50 chance of moving 1 step cast or 1 step west.
Ans: If red is drawn first, replace it before drawing again. If black is drawn, After the step is taken, the move is repeated from the new position and so on
do not replace it. P = 0.64 indefinitely. What is the chance that the particle returns to the origin?
110. In an election, two candidates, Albert and Benjamin, have in a ballot box a and Ans: P = 1/πn for good-sized n
b votes respectively, a > b, for example, 3 and 2. If ballots are randomly drawn 127. As in the two-dimensional walk, a particle starts at an origin 0 in three-space.
and tallied, what is the chance that at least once after the first tally the Think of the origin as centered in a cube 2 units on a side. One move in this
candidates have the same number of tallies? walk sends the particle with equal likelihood to one of the eight corners of the
Ans: P(tie) = 2b/(a + b) or 2/(r + 1) where r = a/b cube. Thus, at every move the particle has a 50-50 chance of moving one unit
111. Marvin gets off work at random times between 3 and 5 P.M. His mother lives up or down, one unit east or west, and one unit north or south. If the walk
uptown, his girl friend downtown. He takes the first subway that comes in either continues forever, find the fraction of particles that return to the origin.
direction and eats dinner with the one he is first delivered to. His mother Ans: 0.239
complains that he never comes to see her, but he says she has a 50-50 chance. 128. A table of infinite expanse has inscribed on it a set of parallel lines spaced 2a
He has had dinner with her twice in the last 20 working days. Explain. units apart. A needle of length 2l (smaller than 2a) is twirled and tossed on the
Ans: Downtown trains run past Marvin's stop at, say, 3:00, 3:10, 3:20, …, table. What is the probability that when it comes to rest it crosses a line?
etc., and uptown trains at 3 :01, 3:11, 3:21, … . To go uptown Marvin must Ans: 2l/πa
arrive in the 1-minute interval between a downtown and an uptown train. 129. Suppose we toss a needle of length 2l (less than 1) on a grid with both
112. Duels in the town of Discretion are rarely fatal. There, each contestant comes at horizontal and vertical rulings spaced one unit apart. What is the mean number
a random moment between 5 A.M. and 6 A.M. on the appointed day and leaves of lines the needle crosses? (I have dropped 2a for the spacing because we
exactly 5 minutes later, honor served, unless his opponent arrives within the might as well think of the length of the needle as measured in units of spacing)
time interval and then they fight. What fraction of due ls lead to violence? Ans: 4/π ≈ 1.27
Ans: 1/6 Mathematical Bafflers
113. The king’s minter boxes his coins 100 to a box. In each box he puts 1 false coin. 130. A gambler devised a game to be played with a friend. He bet ½ the money in his
The king suspects the minter and from each of 100 boxes draws a random coin pocket on the toss of a coin; heads he won, tails he lost. The coin was tossed,
and has it tested. What is the chance the minter's peculations go undetected? and the money handed over. The offer was repeated, and the game continued.
Ans: 0.366 Each time he bet was for ½ the money then in his possession. Eventually the
114. Airborne spores produce tiny mold colonies on gelatin plates in a laboratory. number of times he lost was equal to the number of times he won. Quickly now!
The many plates average 3 colonies per plate. What fraction of plates has Did he gain, lose, or break even?
exactly 3 colonies? If the average is a large integer m, what fraction of plates Ans. He lost, even if they played only twice, or four times, or six.
has exactly m colonies? 131. A prisoner is given 10 white balls, 10 black balls and two boxes. He is told that
Ans: 0.224, 0.4/√𝒎 an executioner will draw one ball from one of the two boxes. If it is white, the
prisoner will go free; if it is black, he will die. How should the prisoner arrange
the balls in the boxes to give himself the best chance for survival?
PROBABILITY AND STATISTICS
Ans: If the prisoner places one white ball in one box and the remaining balls in 145. A long shot poker player draws two cards to the five and six of diamonds and
the other box, his chance of survival would be 0.737 the joker. What are his chances of coming up with a pat hand? (straight or
132. On a certain day, our parking lot contains 999 cars, no two of which have the flush).
same 3-digit license number. After 5:00 p.m. what is the probability that the Ans 0.168
license numbers of the first 4 cars to leave the parking lot are in increasing 146. In Puevigi, the game of craps is played with a referee calling the point by adding
order of magnitude. together the six faces (three on each die) visible from his vantage point. What is
Ans: 1/24 the probability of making 16 the hard way? (That is, by throwing two eights.)
133. A hospital nursery contains only two baby boys; the girls have not yet been Ans: zero
counted. At 2:00 p.m. a new baby is added to the nursery. A baby is then 147. Max and his wife Min each toss a pair of dice to determine where they will
selected at a random to be the first to have its footprint taken. It turns out to be a spend their vacation. If either of Min’s dice displays the same number of spots
boy. What is the probability that the last addition to the nursery was girl? as either of Max’s, she wins and they go to Bermuda. Otherwise, they go to
Ans: 2/5 Yellowstone. What is the chance they’ll see “Old Faithful” this year?
134. Assume that a single depth charge has a probability of ½ of sinking a Ans: 0.514
submarine, ¼ of damage and ¼ of missing. Assume also that two damaging 148. There are four volumes of an encyclopedia on a shelf, each volume containing
explosions sink the sub. What is the probability that 4 depth charges will sink 300 pages, (that is, numbered 1 to 600), but these have been placed on the
the sub? shelf in random order. A bookworm starts at the first page of Vol. 1 and eats his
Ans: 251/246 way through to the last page of Vol. 4. What is the expected number of pages
135. If 2 marbles are removed at a random from a bag containing black and white (excluding covers) he has eaten through?
marbles, the chance that they are both white is ⅓. If 3 are removed at random, Ans: 500
the chance that they all are white is 1/6. How many marbles are there of each 149. Venusian batfish come in three sexes, which are indistinguishable (except by
color? Venusian batfish). How many live specimens must our astronauts bring home in
Ans: 6 white and 4 black order for the odds to favor the presence of a “mated triple” with its promise of
136. An expert gives team A only a 40% chance to win the World Series. Basing his more little batfish to come?
calculation on this gambler offers 6 to 5 odds on team B to win the first game. Is Ans. With four specimens, the odds in favour of a mated triple are only
his judgement sound? 4/9. But if payload limitation permit five to travel to Earth, the odds go up
Ans: the gambler is on the safe side, P = 0.5455 to 50/81.
137. A salesman visits ten cities arranged in the form of a circle, spending a day in 150. In the final seconds of the game, your favorite NΒΑ team is behind 117 to 118.
each. He proceeds clockwise from one city to the next, except whenever leaving Your center attempts a shot and is fouled for the 2 nd time in the last 2 minutes
the tenth city he may go to either the first or jump to the second city. How many as the buzzer sounds. Three to make two in the penalty situation. Optimistic?
days must elapse before his location is completely indeterminate, i.e., when he Note: the center is only a 50% free-thrower. What are your team’s overall
could be in any one of the ten cities? chances of winning?
Ans: 83 days Ans: 11/16 (about 69%)
138. Three dart players threw simultaneously at a tic-tac-toe board, each hitting a 151. One of a pair of dice is loaded so that the chance of a 1 turning up is 1/5, the
different square. What is the probability that the three hits constituted a win at other faces being equally likely. Its mate is loaded so that the chance of a 6
tic-tac-toe? turning up 1/5, the other faces being equally likely. How much does this loading
Ans: 2/21 increase the probability of throwing a 7 with the two dice?
139. All the members of a fraternity play basketball while all but one play ice hockey; Ans: only one part in 750
yet the number of possible basketball teams (5 members) is the same as the 152. If all 720 permutations of the digits 1 through 6 are arranged in numerical order,
number of possible ice hockey teams (6 members). Assuming there are enough what is the 417th term?
members to form either type of team, how many are in the fraternity? Ans 432516
Ans: 15 members 153. The local weather forecaster says “no rain” and his record is 2/3 accuracy of
140. A game of super-dominoes is played with pieces divided into three cells instead prediction. But the Federal Meteorological Service predicts rain and their record
of the usual two, containing all combinations from triple blank to triple six, with is ¾. With no other data available, what is the chance of rain?
no duplications. For example the set does not include both 1 2 3 and 3 2 1 since Ans: 3/5
these are merely reversals of each other. (But, it does contain 1 3 2.) How many 154. A sharp operator makes the following deal. A player is to toss a coin and
pieces are there in a set? receive 1, 4, 9, … n2 dollars if the first head comes up on the first, second, third,
Ans: 196 … nth toss. The sucker pays ten dollars for this. How much can the operator
141. Martian coins are 3-sided (heads, tails, and torsos), each side coming up with expect to make if this is repeated a great many times?
equal probability. Three Martians decided to go odd-man-out to determine who Ans: four dollars per game
pays a dinner check. (If two coins come up the same and one different, the 155. In 1969, the World Series will begin in the stadium of the American League
owner of the latter coin foot the bill). What is the expected number of throws pennant winner. Assume the contenders are evenly matched. What is the
needed in order to determine a loser? probability that the series will end where it began?
Ans: 1½ Ans: 5/8
142. There are three families, each with two sons and two daughters. In how many 156. In a carnival game 5 balls are tossed into a square box divided into 4 square
ways can all these young people be married? cells, with baffles to ensure that every ball has an equal chance of going in any
Ans: 80 cell. The player pays $1 and receives $1 for every cell which is empty after the 5
143. How many three digit telephone area codes are possible given that: (a) the first balls are thrown. How much does the operator expect to make per game?
digit must not be zero or one; (b) the second digit must be zero or one; (c) the Ans: a nickel a game
third digit must not be zero; (d) the third digit may be one only if the second digit 157. Having lost a checker game, a specialist in learning programs threw one of the
is zero. red checkers out the window. His wife reboxed the 12 black pieces and 11 red
Ans :136 possible codes pieces one at a time in random fashion. The number of black checkers in the
144. Six men decide to play Russian roulette with a six gun loaded with one box always exceeded the number of reds. What was the a priori probability of
cartridge. They draw for position, and afterwards, the sixth man casually this occurrence?
suggests that instead of letting the chamber rotate in sequence, each man spin Ans: 1/23
the chamber before shooting. How would this improve his chances? GOD BLESS
Ans. 0.1
1. Determine is the stiffness factor of a pavement if its modulus of 14. Given the following cross-section notes for a road grading work:
elasticity is 180 MPa and whose subgrade modulus is 40 MPa. −3.2 1.2 1.2 1.8
A. 0.60 𝑥1 0 3.5 𝑥2
2. A -6% grade and a +2% grade intersect at STA 12 + 200 whose The road bed is 9 m wide and the side slope for cut is 1:1 and for fill
elevation is at 25.632 m. The two grades are to be connected by a is 1.5:1. Determine the area of cut of the section.
parabolic curve; 160 m long. What is the elevation of the first quarter A. 9.4
point on the curve? 15. A simple curve is to connect two tangents whose angle of intersection
A. 28.432 is 30°. The curve is to pass a point B whose perpendicular distance
3. A steel tape is 100 m long at a standard pull of 65 N. Compute the from the tangent through PC (at point C. is 9.15 m. Point C is 27.45 m
pull correction in mm if during measurement the applied pull is 40 N. from the point of intersection of the tangents. Determine the length
The tape has a cross-sectional area of 3.18 mm² and a modulus of of the curve.
elasticity E = 200 GPa. A. 226.54 meters
A. -3.93 16. An Engineer’s level uses a level tube with radius of curvature of 4 m.
4. A rectangular field was measured using a 100-m tape which was If during observation, the bubble is off-centered through 4 spaces,
actually 10-cm too short. The recorded area was 4500 m². What is the what error on observed vertical distance on a station 120 m away if
true area of the field? one space on the tube is 0.5 mm long?
A. 4491 m² A. 0.06 meters
5. A line on a map was drawn at a scale of 5:100,000. If a line in the map 17. A spiral easement curve has a length of 100 m with a central curve
is 290 mm long, the actual length of the line is: having a radius of 300 m. Determine the offset distance from the
A. 5.8 km tangent to the third-quarter point of the spiral.
6. A student recorded the following repeated paces for a given line. A. 2.34
456, 448, 462, 447, 452, 455 18. The cross-sectional area of station 2 + 210 is 40 square meters in fill
If his pace factor is 0.628 m/pace, what is the approximate length of and at station 2 + 810 is 60 square meter in cut. The free haul distance
the line in meters? is 100 meters. The balancing point is at Sta. 2 + 510. The ground
A. 285 m surface is sloping uniformly upward from Sta. 2 +210 to Sta. 2 + 510
7. The length intercepted on the stadia rod is 2.83 m and the line of and also uniformly upward from Sta. 2 + 510 to Sta. 2 + 810.
sight makes an angle of 4°30’ with the horizontal. Find the vertical Determine the stationing (along fill) of the limits of free haul.
distance, in m, from the center of the instrument to the rod, if the A. 2 + 454.95
stadia constant is 0.3 m and the stadia interval factor is 100. 19. A closed triangular traverse has the following data:
A. 22.16 m Line Bearing Distance
8. A line was measured with 20-m tape. There were 3 tallies and 6 pins, AB N 60° E 1000 m
and the distance from the last pin and the end of the line was 3.75 m. BC Due South __________
Determine the length of the line in meters. CA N 60° W __________
A. 723.75 An area of 320,000 square meter is cut-off starting from corner A to
9. In 1985, the magnetic bearing of line AB was N 16° 40’ W and the point F on line BC. What is the length of line AF?
magnetic declination at that time was 1° 15’ E. If the secular variation A. 898.4 m
per year is 3’ E, what would be the magnetic bearing of the line in 20. During peak hours, 4400 vehicles pass through a certain highway
1998? from 9:00 am to 11:00 am, with space mean speed of 20 kph. What is
A. N 17° 19’ W the traffic density in vehicles per kilometer?
10. Point P is between points Q and R. The distances of Q and R from A. 110
point P are 1000 m and 2000 m, respectively. Measured from point P, 21. In the two-peg test of a dumpy level, the following observations were
the angle of elevation of point R is 8°30’, while that of point Q is θ. taken:
The difference in the elevations of Q and R is 44.4 m, with R being Rod Reading on A Rod Reading on B
lower than Q. Considering the effects of curvature and refraction, the Instrument at A 1.506 2.024
value of θ is nearest to: Instrument at B 0.938 1.449
A. 18°58’ What is the correct difference in elevation between A and B?
11. Elevation of triangulation station A is 250 while that of B is 685 m. In A. 0.5145 m
between stations A and B is a mountain C with elevation 325 m. The 22. TP1 has an elevation of 115.15 m and the foresight taken at the point
height of transit placed at A is 1.2 m. The distances AC is 30 km and is 0.30 m. If the BS taken at BM is 1.42, the elevation of BM is:
BC is 50 km. Determine the height of the tower that can be A. 114.03 m
constructed at B such that the line of sight will just pass through the 23. Due to maladjustment, a transit, with the telescope in normal
mountain C with a clearance of 1.5 m. position, is deflected 15” to the left of its correct position, or not
A. 37 meters perpendicular to the horizontal axis. Determine the error in the
12. The cross-sectional area of a road with width of 10 m is 42.9 square measured horizontal angle if the vertical angle of the first point is 46°
meters. The cross-sectional area is as follows: and that of the second point is 74°.
9.8 0 7.4 A. 32.82”
2.4 𝑥 1.2 24. A 100-m tape weighing 5.08 kg was used to measure a line. It was
Determine the value of x. supported at the end points, midpoint, and quarter point and the
A. 3.94 meters tension applied is 60 Newtons. If the total measured distance is
13. A surveying instructor sent out six groups of students to measure a 2345.76 m, what is the correct distance of the line?
distance between two points on the ground. The students came up A. 2341.55
with the following six different values: 360.52, 360.35, 360.89, 361.02, 25. A simple curve is to be designed for a maximum speed of 90 kph. The
359.98, and 360.78 meters. Assuming these values are equally reliable coefficient of friction between the tires and the pavement is known
and that the variations result from accidental error, determine the to be 0.4. If the super elevation is to be limited to 12%, what should
most probable value of the distance measured. be the degree of curvature? Use arc basis.
A. 360.59 meters A. 9.34
26. A descending grade of 4.2% intersects an ascending grade of 3% at 40. The length intercepted on the stadia rod is 2.83 m. and the line of
station 12 + 325 at elevation 14.2 m. These two grades are to be sight makes an angle of 4°30’ with the horizontal. Find the vertical
connected by a 260-m vertical parabolic curve. A reinforced concrete distance in m., from the center of the instrument to the rod, if the
culvert pipe with overall diameter of 105 cm is to be constructed with stadia constant is 0.30 and the stadia interval factor is 100.
its top 30 cm below the subgrade. What will be the invert elevation? A. 22.16 m
A. 15.13 m 41. A test vehicle moving at 40 kph was stopped by applying the brakes
27. During peak hours, 3800 vehicles pass through a certain highway and the length of the skid mark was 12.2 m. If the average skid
from 9:00 am to 11:00 am, with space mean speed of 20 kph. What is resistance of the level pavement is known to be 0.70, determine the
the traffic density in vehicles per kilometer? brake efficiency of the test vehicle.
A. 95 A. 73.7%
28. If 2340 vehicles per hour passes a certain lane of road with average 42. The area bounded by the waterline of a reservoir and the contours at
speed of 52 kph, find the appropriate spacing of these vehicles. an interval of 1.7 m are as follows: A1 = 15,430 m²; A2 = 12,980 m², A3
A. 22.2 m = 10,650 m², A4 = 8,540 m², A5 = 5,270 m², and A6 = 2,180 m².
29. The following data were taken on five cars traversing a 1.5-km Calculate the volume of the reservoir by prismoidal formula in cu. m.
highway. A. 78,911
Car Time (minutes) 43. The length of the spiral curve is 82 m. and the radius of the central
A 1.3 curve of the spiral curve is 260 m. Compute the length of throw.
B 1.1 A. 1.08
C 1.4 44. Find the area of a piece of land with an irregular boundary as follows:
D 1.0 STA Offset Distance (m)
0 + 000 5.59
E 1.2
0 + 015 3.38
Calculate the space mean speed.
0 + 030 2.30
A. 75 kph
0 + 045 3.96
30. The ground makes a uniform slope of 5.2% from STA 8 + 890 to
0 + 060 4.80
Station B. At STA 8 + 890, the center height of the roadway is 4.13 m
The stations are on straight-line boundary. Find the area of the land
fill. At the other station, the center height is 7.57 m cut. If the finish
in m² by Trapezoidal Rule.
road has a uniform grade of -2.6%, what is the stationing of B?
A. 222.53
A. 9 + 040
45. The DMD of the line BC is equal to 426.20. The departure of the
31. A particular station has the following earthwork cross-section
9.0 4.5 4.5 𝑥 previous line AB is 117.56. What is then the departure of the line BC?
𝑦 +4.0
+2.0 A. 191.08
+2.0 +1.0
If the width of the road base is 9 m, and the side slopes are 1V:1.5H, 46. A closed traverse has the following data:
the quantity of x is equal to: Line Distance Bearing
A. 6 AB 179.00 N 47° 02’ 14” E
32. Using arc basis, a 3.2-degree curve with central angle of 18° has an BC 258.20 S 69° 35’ 59” E
external distance of: CD ___________ S 39° 35’ 17” W
A. 4.46 m DE ___________ S 87° 29’ 48” W
33. The offset distance from PC to PT of a simple curve is 18 m. The angle EA 145.41 N 24° 48’ 09” W
of intersection of the tangents is 24°. If the stationing of PT is 45 + What is the length of line CD?
158.32, what is the stationing of PI? A. 202.4 m
A. 45 + 115.36 47. A 12-degree simple curve is to be designed for a maximum speed of
34. A car moving at 80 kph on a level road suddenly sees an obstruction 88 kph. The coefficient of friction between the tires and the pavement
76 m ahead. If the perception-reaction time is 0.5 second and the is 0.4. What is the required super elevation in percent?
coefficient of friction between tires and the pavement is 0.4, how far A. 18.7
from the obstruction will it stop? 48. The areas in cut of two irregular sections 65 m apart are 36 square
A. 2 m meter and 72 square meters, respectively. Base width is 10 m, side
35. A spiral 80 m long connects a tangent with a 6.5° circular curve. slope is 3H:2V. Using the prismoidal correction formula, find the
Determine the spiral angle at the first quarter point. corrected volume of cut, in cubic meter between the two stations.
A. 0.27° A. 3,459.5
36. An 80-m spiral connects a tangent with a 180-m radius circular curve. 49. A spiral easement curve has a length 100 m. with a central curve
The maximum velocity in kph that a car could pass through the curve having a radius of 300 m. Determine the degree of spiral at the third
without skidding is nearest to: quarter point.
A. 73.68 A. 3.82
37. The long chord of a compound curve is equal to 250 meters and the 50. A car moving at 80 kph on a level road suddenly sees an obstruction
angles it makes with the tangents equal to 8° and 10°, respectively. 76 m. ahead. If the perception-reaction time is 0.5 sec. and the
Find the radii, R1 and R2 when the common tangent is parallel to the coefficient of friction between the tires and the pavement is 0.4, how
long chord. far from the obstruction will it stop?
A. 998.33 m & 639.54 m A. 1.98 m
38. An 80 m. spiral connects a tangent with a 180 m. radius circular curve. 51. The area bounded by the waterline of a reservoir and the contours at
What is the max. velocity in kph that a car could pass through the an interval of 1.7 m. are as follows:
curve without skidding? A1 = 15430 m² A2 = 12980 m² A3 = 10650 m²
A. 73.68 kph A4 = 8540 m² A5 = 5270 m² A6 = 2180 m²
39. A vertical summit curve has tangent grades of + 5% and – 3.8%. The Calculate the volume of the reservoir by prismoidal formula in cu.m.
horizontal distance from the P.C. to the highest point of the curve is A. 78911
113.64 m. Find the total length of the curve.
A. 200
52. A sight is taken with an engineer’s level at rod held 100 m. away, and 65. The slope distance and vertical angle between points A and B were
an initial reading of 1.83 m. was observed. The bubble is then leveled measured with a total-station instrument as 9585.26 ft and 8°17’40”,
through 6 spaces on level tube and the rod reading is 1.91 m. What respectively. The hi and rod reading were equal. If the elevation of A
is the sensitivity of the bubble tube in seconds of arc? is 1238.42 ft above the mean sea level, compute the elevation of B.
A. 27.5 A. 2624.08 ft
53. A vein has a strike of N. 10°15’W. and a dip of 43°40’W. A drift in the 66. Assume the magnetic bearing line AB read in 1878 was N26°15’E. The
vein has a grade of 2%. Find the bearing of the drift. declination at that time and place was 7°15’W. In 1984 the declination
A. N11°27’W was 4°30’E. What is the needed magnetic bearing in 1984?
54. The radius of the horizontal curve is 280 m. long having a super A. N14°30’E
elevation of 0.06. How fast can a car move along the curve in kph if it 67. A surveyor sets up a transit at point P which is at the inner portion of
has a coefficient of lateral friction equal to 0.12. a four-sided tract of land ABCD and read the bearings and measures
A. 77.78 kph the distances as follows:
55. Applying full brakes at a speed of 60 kph, the car traveled 40 m. until Line Bearing Distance
it stopped. Determine the average skid resistance. PA N. 40°30’ W 420.35
A. 0.35 PB N. 38°00’ E 530.15
56. The perpendicular distance between the two parallel tangents of a PC S. 70°00’ E 480.75
reversed curve is 8 m. and the chord distance from the PC to the PT PD S. 60°15’ W 695.10
is equal to 30 m. Compute the central angle of the reverse curve. What is the area of the track in hectares?
A. 30°55’ A. 50.14
57. A vertical summit curve has its highest point of the curve at a distance
68. An engineer’s level uses a level tube with a radius of curvature of 4
of 48 m. from the P.T. The back tangent has a grade of + 6% and a m. If during observation, the bubble is off-centered though 4 spaces,
forward grade of -4%. If the stationing of the PT is 10 + 100, what error on observed vertical distance on a station 120 meters away
determine the length of vertical summit curve in meters. if one space on the tube is 0.5 mm long?
A. 120 A. 0.06 m
58. The offset distance of the simple curve from the P.T. to the tangent 69. A car was traveling at a speed of 80 kph. The driver saw a roadblock
line passing thru the P.C. is equal to 120.20 m. The simple curve has
80 m. ahead and stepped on the brake causing the car to decelerate
an angle of intersection of 50°. Find the radius of the simple curve. uniformly at 10 m/sec2. Assuming perception-reaction time is 2 sec.,
A. 336.49
determine the distance from the roadblock to the point where the car
59. Find the area of a piece of land with an irregular boundary as follows: stopped in meters.
Stations Offset Distances A. 10.87 m
0 + 000 5.59 70. A compound curve has a common tangent 84.5 m. long, which makes
0 + 015 3.38 angles of 16° and 20° with the tangents of the first and the curves
0 + 030 2.30 respectively. If the length of the tangent of the first curve is 38.6 m.
0 + 045 3.96 Determine the radius of the second curve.
0 + 060 4.8 A. 260.3
60. The stations are on a straight-line boundary. Find the area of the land 71. The spiral easement curve has a length of spiral equal to 80 m. and
using trapezoidal rule. the radius of the central curve of the spiral curve is 192.84 m.
A. 222.525 m² Compute the deflection angle at the end point of the spiral.
61. The cross-sectional area of station 2 + 210 is 40 m² in fill and at A. 3.96°
station 2 + 810 is 60 m² in cut. The free haul distance is 100 m. The 72. Suppose four measurements of a distance are recorded as 482.16,
balancing point is at station 2 + 510. The ground surface is sloping 482.17, 482.20, and 482.18 and given weights of 1, 2, 2, and 4,
upward from station 2 + 210 to station 2 + 510 and also uniformly respectively, by the survey-party chief. Find the weighted mean.
upward from station 2 + 510 to station 2 + 810. Determine the A. 482.18 feet
stationing (along fill) of the limits of free haul. 73. Compute the volume of excavation between station 24 + 00, with an
A. 2 + 454.95 end area of 711 ft2, and station 25 + 00, with an end area of 515 ft2.
62. The ground makes a uniform slope of 5.2% from sta. 8 + 890 to A. 2268 yd3
station B. At sta. 8 + 890, the center height of the roadway is 4.13 m. 74. On a vertical photograph, the length of an airport runway measures
fill. At the other station, the center height is 7.57 m. cut. If the finish 4.24 in. On a map plotted to a scale of 1:9600 it extends 7.92 in. What
road has a uniform grade of – 2.6%, what is the sta. of B? is the photo scale at the runway elevation?
A. 9 + 040 A. 1490 feet
63. The offset distance from PC to PT of a simple curve is 18 m. and the 75. In order to measure the height of a mountain, a surveyor takes two
angle of intersection of the tangents is 24°. If the stationing of P.T. is sightings from a transit 1½ m high. The sightings are taken 1200 m
45 + 158.32, what is the stationing of the P.I.? apart from the same ground elevation. The first measured angle of
A. 45 + 115.30 elevation is 51°, and the second is 38°. To the nearest meter, what is
64. Assume the measured angles of a certain triangle are A = 49°51’15”, the height of the mountain (above the ground elevation).
wt 1; B” = 60°32’08, wt 2; and C = 69°36’33”, wt 3. Perform a weighted A. 2554 m
adjustment of the angles.
A. 11x = 4” and x = +0.36”
 SURVEYING PRACTICE PROBLEMS
1. From the given horizontal distance measurement notes using steel chains, 10. In this example of a start-up of a project, two field engineers are measuring distances
determine the discrepancy ratio of line 10-11. between control points (CP) that had been set by a professional surveyor. The
TAPE FORWARD BACK MEAN distance they measured and recorded, between CP 1004 and CP 1005, was found
LINES DIST. FT. to be 500.17. Upon checking the "report of survey" from the professional surveyor,
10 - 11 105.85’ 105.91’ 105.88’ the distance between CP 1004 and CP 1005 is supposed to be 500.00'. They aren’t
concerned though because they haven't made the necessary corrections. At the time
11-19 138.50’ 138.53’ 138.51’ of the measurement, the temperature of the chain was a scorching 103 F. The
difference in elevation between the two control points is 20 feet. The calibrated
19-15 168.29 168.25’ 168.27’ length of chain is 100.02'. What is the actual distance between the control points that
must be laid out in the field to obtain the true distance between the control points?
Ans. 499.97
15-12 185.16’ 185.18’ 185.17’
11. A 100-foot steel chain (known to be 99.93 feet) was used to measure between two
building points. A distance of 147.44 feet was recorded at a temperature of 43°F.
10-21 91.24’ 91.28’ 91.26’ What is the distance after correcting for temperature and chain length error?
Ans. 147.31
Hint: 12. The slope distance between the two points is 24.776 feet and the slope angle is
Forward – Back = Discrepancy 1°17’. Compute the horizontal distance
𝐷𝑖𝑠𝑐𝑟𝑒𝑝𝑎𝑛𝑐𝑦 Ans. 24.770
Discrepancy Ratio =
𝑀𝑒𝑎𝑛 13. The slope distance between two points is 42.71 feet, and the difference in elevation
Ans. 1/1764 between them is 3.56 feet. Compute the horizontal distance.
2. From the given traverse chaining field notes, determine the discrepancy ratio of each Ans. 24.52 ft
line. 14. A distance of 328 feet was measured along a 2% slope. Compute the horizontal
LINE FORWARD BACK MEAN distance.
DIST. FT. Ans. 327.93
6-4B 157.86’ 157.87’ 157.865’ 15. It is required to lay out a rectangular building, 75 feet wide by 100 feet long. If the
100’ steel chain being used is 99.94 feet long, what distances should be laid out?
4B-4 146.80’ 146.84’ 146.82’ Ans. 75.045’ x 100.06’
16. A concrete slab measuring 10 feet by 85 feet is to be laid out by a chain known to
4-4C 142.70’ 142.71’ 142.705’ be 100.07 feet long under standard conditions. What distances should be laid out?
Ans.= 9.99’ x 84.94’
4C-5 46.89’ 46.89’ 46.89’ 17. A 100’ steel chain standardized at 99.98’ was used to measure a distance between
control points of 1275.36 feet when the field temperature was 87°F. The ground
5-6 136.33’ 136.32’ 136.325 was sloping at 5%. What is this distance under standard conditions?
Ans. 268.20 ft
Ans. . .01, .04, .01, .00, .01 18. A steel chain known to be 99.94 feet under standard conditions is used to measure
3. Suppose a chain was used to lay out centerline points on a two-mile length of the distance between two control points. If a distance of 178.4 feet was recorded at
roadway. After the work was finished, the field engineer decided to check that chain 58°F, what distance should be measured at a temperature of 75°F?
at the local baseline. Upon checking, it was found the chain was actually 100.02 Ans. 178.38
rather than 100.00. What effect will this have ton the two miles of centerline stakes? 19. A steel chain known to be 100.03 feet is used to measure the distance between two
How far will the stake be at the end of the line (10,560 feet away)? building corners. If the distance between the corners is supposed to be 268.33 feet
Ans. 2.11’ and the field temperature is 97°F, then what distance should be laid out?
4. A chain was used to check the distance between two control points on a large Ans. 268.20
building project. The distance was measured and recorded as 600.172 m. After the 20. Two control points are known to be 487.63 feet apart. Using a 200’ chain known to
work was finished, it was decided to check that chain at the location baseline. Upon be 199.96 feet under standard conditions, what distance should be measured when
checking, it was found that the chain was actually 29.992 m rather than 30.000. What the field temperature is 78°F?
effect will this have on the recorded distance? What is the actual length between the Ans. 487.70 ft
two control points?
Ans. 600.002 Situation 1 - Given: Given: I = 44°, R = 400 and PI Sta. = 12 + 72.18. Compute the
5. A 100-foot chain is used to lay out bridge abutments that are 50 feet apart. The following.
ground is sloping at the rate of 10’ per 100 feet. What distance will need to be laid 21. Tangent
out on the slope to maintain the 50’ horizontal spacing? Ans. 161.61’
Ans. 50.25’ 22. Degree of curve. Use Da = 5279.25/R
6. A construction surveyor measured the length between two control monuments with Ans. 14.324°
a 50-meter chain. One point was at the bottom of a hill and the other was on the side 23. Length of curve.
of the hill. A slope measurement was made by holding the zero end of the chain at Ans. 307.18;
the center of the instrument and reading 45.290 meters at the point. A zenith angle 24. Long chord.
of 75° was made to the point. Ans. 299.69
Ans. 43.747 m 25. External Distance
7. On a cold, blustery day, a 100-foot chain is used to lay out building corners for a Ans. 31.41’
250-foot by 400-foot structure. The temperature of the chain at the time of 26. Middle ordinate
measurement is 23° F. What distance will need to be laid out to set the points at the Ans. 29.13
prescribed distances? 27. Stationing of the PC
Ans. 250.075 and 400.120 Ans. 11 + 10.57
8. On a hot, sunny day, a 30-meter chain is used to measure the distance between two 28. Station of the PT.
highway centerline monuments. The temperature of the chain at the time of Ans. 14+ 17.75
measurement is 38°C. The distance was measured and recorded as 499.897 29. Deflection per foot. (Board exam problem)
meters. What is the actual distance? Ans. 0.071620°/ft
Ans. 500.000 m 30. Deflection increment to station 11 + 50.
9. A distance measuring 250 feet to a proposed pier is to be laid out on a bridge project. Ans. 2° 49’ 26”
The terrain from the control point to the pier is sloping at a 7% grade down into the 31. Deflection increments for the 50-foot station intervals
river valley. The temperature of the chain at the time of the measurement is 45°F. Ans. 3° 34’ 52”
The chain used is known to be 100.01 when compared to a standard at 68°F. What 32. Total deflections at sta (a) 11+50, (b) 12+00, (c) 12+50 and (d) 13+00.
is the distance that must be laid out in the field to obtain the desired distance to the Ans. (a) 2° 49’ 26” (b) 6° 24’ 18”, (c) 9° 59’ 09” (d) 13° 34’ 01”
pier? 33. Short chords at (a) 11+50, (b) for 50-foot intervals and © for the closing arc into the
Ans. 250.613 PT.
Ans. (a) 39.414’ (b) 49.967’ and (c) 17.748
34. Long chords at sta (a) 11+50, (b) 12+00, (c) 12+50 and (d) 13+00.
Ans. (a) 39.41’, (b) 89.244’, (c) 138.725’ and (d) 187.665’
 SURVEYING PRACTICE PROBLEMS

35. Given I = 44°, R = 400’ and PI Sta. = 12 + 72.18. Calculate the 50-foot offset long 49. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
chord to 12 + 50. gravity) of an asphalt concrete is 2.550. the components are specified as follows:
Station Deflection Total Short Chord Long Chord 50’ Offset 50’ Offset Material Specific Apparent specific By weight
Increment Deflection short Chord Long gravity gravity %
Chord
Asphalt cement 1.020 - 6.3
14 + 17.75 PT 1° 16’ 16” 22° 00’ 00” 17.75’ 299.68 19.96’
14 + 00 3° 34’ 52” 20° 43’ 44” 49.97’ 283.16 56.22’
Limestone dust 2.820 2.650 13.7
13 + 50 3° 34’ 52” 17° 08’ 52” 49.97’ 235.87’ 56.22’ Sand 2.650 2.905 30.4
13 + 00 3° 34’ 52” 13° 34’ 01” 49.97’ 187.66 56.22’ Gravel 2.650 2.873 49.6
12 + 50 3° 34’ 52” 9° 59’ 09” 49.97’ 138.72’ 56.22’ What is the asphalt absorption?
12 + 00 3° 34’ 52” 6° 24’ 17” 49.97’ 89.24’ 56.22’ Ans. 2.18%
11 + 50 3° 34’ 52” 2° 49’ 26” 39.41’ 39.41’ 44.34’ 50. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
11 + 10.57 PC 3° 34’ 52” 0° 0 0 0 gravity) of an asphalt concrete is 2.550. the components are specified as follows:
Ans. 156.06’ Material Specific Apparent specific By weight
36. The following is a road centerline from PVI 1 through PVI 2 to PVI 3. The first step gravity gravity %
in vertical curve calculations is to determine the gradient between the PVl’s. Asphalt cement 1.020 - 6.3
Calculate the gradient from PVI 1 station 21 + 50, elevation 626.34; to PVI 2, station Limestone dust 2.820 2.650 13.7
26+00, elevation 655.06; and from PVI 2 to PVI 3 station 29+25, elevation 643.54. Sand 2.650 2.905 30.4
Ans. 6.38% and -3.54% Gravel 2.650 2.873 49.6
How much would the percent air voids of the mixture change if the weight of the
Situation 2 - This information is taken from a plan:
asphalt were increased by 2%
PVI = 26+00 Elevation = 655.06
Ans. 0.14 decrease
g₁ = +6.38% g₂ = -3.54%
51. Given the end areas below, calculate the volumes of cut and fill between stations
L = 4 stations
351 + 000 and 352 + 50. If the material shrinks 12 percent, how much excess cut or
37. Compute the rate of change of the gradient in percent per station.
fill is there?
Ans. 1.24
End areas, m3
38. Compute the vertical distance from PVI to curve.
Ans. 4.96 Station Cut Fill
39. Compute the vertical distance from Station 24 + 50 to the curve. 351 + 00 57.93
Ans. 0.31 351 + 50 52.28
351 + 75 0 23.58
Station 3 – Given: 352 + 00 8.40 3.73
PVISTA = 44+00 PVIelev = 686.45 g₁ = -3.34% g₂ = +1.23 352 + 14 13.80 0
L₁ = 600 feet L₂ = 400 feet 352 + 50 33.34
40. Calculate the station of (a) BVC, PVIA, PVIB and the EVC. Ans. 3117.4 m3
Ans BVC = PVI – L₁ = 44+00 - 600 = 38+00 52. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
EVC = PVI + L₂ = 44+00 + 400 = 48+00 gravity) of an asphalt concrete is 2.550. the components are specified as follows:
PVIA = PVI – L₁/2 = 44+00 - 600/2 = 41+00 Material Specific Apparent By weight %
PVIB = PVI + L₂/2 = 44+00 + 400/2 = 46+00 gravity specific gravity
41. Calculate the elevation of (a) BVC, PVIA, PVIB and the EVC. Asphalt cement 1.020 - 6.3
Ans. BVC = 686.45 + [(3.34)(6)] = 706.49 Limestone dust 2.820 2.650 13.7
PVIA = 686.45 + [(3.34)(3)] = 696.47 Sand 2.650 2.905 30.4
PVIB = 686.45 + [(1.23)(2)] = 688.91 Gravel 2.650 2.873 49.6
EVC = 686.45 + [(1.23)(4)] = 691.37 What is the air void content if the bulk specific gravity of the mixture is 2.340?
42. Calculate the gradient gAB between PVA and PVB Ans. 8.2%
Ans. -1.51% 53. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
43. Calculate the elevation of the CVC. Working from PVIA. gravity) of an asphalt concrete is 2.550. the components are specified as follows:
Ans. 691.94 Material Specific Apparent By weight %
44. Calculate the rate of change of the gradient for curve 1 and curve 2. gravity specific gravity
Ans. -0.1525 and -0.3425 Asphalt cement 1.020 - 6.3
Limestone dust 2.820 2.650 13.7
45. Calculate the low point elevation and elevation given Sand 2.650 2.905 30.4
g₁ = -2%, g₂ = 1.6%, L = 8 statinos and PVI @87+00 PVI Elev 743.00 Gravel 2.650 2.873 49.6
Ans. 87 + 44.44 and 746.55
What is the bulk specific gravity of the aggregate?
46. Calcuate the area of an irregular shape with interval between offet lines x = 10 and
Ans. 2.67
offset measurements from the baseline to the limit of the irregular shape equal to
54. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
2.0, 5.1, 6.0, 8.3, 10.5, 10.1, 7.7, 3.1, 0.0, respectively.
gravity) of an asphalt concrete is 2.550. the components are specified as follows:
Ans. 518 sq. ft.
Material Specific Apparent By weight
47. Calculate the volume using the method of average end-area for one section of
gravity specific gravity %
roadway given: Area 1 = 153 sq. ft. at 21 + 0 and Area 2 = 250 sq. ft. at 20 + 0.
Asphalt cement 1.020 - 6.3
Ans. 746.3 cu. Yd.
48. Determine the volume of the depression that will be filled from the contour map with Limestone dust 2.820 2.650 13.7
the following data. The contour interval of 2 meters. Sand 2.650 2.905 30.4
Gravel 2.650 2.873 49.6
Contour (meters) Area of the contour (m²) What is the effective specific gravity of the aggregate?
122 125 Ans. 2.84
55. The specific gravity of a voidless mixture (i.e., the maximum theoretical specific
gravity) of an asphalt concrete is 2.550. the components are specified as follows:
120 77
Material Specific Apparent specific By weight %
gravity gravity
118 32
Asphalt cement 1.020 - 6.3
Limestone dust 2.820 2.650 13.7
116 20
Sand 2.650 2.905 30.4
Gravel 2.650 2.873 49.6
114 10
What is the apparent specific gravity of the aggregate?
Ans. 393 Ans. 2.85
TRIGONOMETRY
1. Find the area of the triangle having sides 6 mi, 10 mi, 14 mi. 27. Using fundamental identities, write the following expression in terms of sines and cosines,
Ans. 25.98 sq mi and then simplify:
2. On a circle of radius 30 in, the length of the arc intercepted by a central angle of rad is tan x - cot x
Ans. 10 tan x + cot x
3. On the same circle a central angle of 50intercepts an arc of length Write the final answer in terms of cosine function.
Ans. 25π/3 in Ans. 1 – 2 cos2 x
4. On the same circle an arc of length 1½ ft subtends a central angle 28. Find the complement of 32°.
Ans. ⅗ rad Ans. 58°
5. For a circle of radius 30 in, the area of a sector intercepted by a central angle of ⅓ rad is 29. Find the radian of a central angle that subtends an arc of length from 2.5 in from a circle of
Ans. 150 in² radius 2 in.
6. For a circle of radius 18 cm, the area of a sector intercepted by a central angle of 50° is Ans. 1.25 in
Ans. 45π or 141 cm² 30. Find cos θ if sin θ = 0.45 and θ is in the second quadrant.
7. A bicycle with 20-in wheels is traveling down a road at 15 mi/h. Find the angular velocity of Ans. -0.8930
the wheel in revolutions per minute. 31. A man who is 6 feet tall is standing 10 feet from the base of a lamppost. The man’s shadow
Ans. 252 rev/min has a length of 4 feet. How tall is the lamppost?
8. Solve for sin x, if tan 3x = 5 tan x. Ans. 21 ft
Ans. 0.3536 32. In a 25-inch television set, the length of the screen’s diagonal is 25 in. If the screen’s height
9. If sin θ = k, which of the following is not correct: is 15 in, what is its width?
Ans. 20 in
Ans. sec θ = 1/√𝑘 2 − 1
33. Given that sin 34° = 0.5592, find an acute angle θ such that cos θ = 0.5592.
10. Given the following relations:
Ans. 56°
r cos A cos B = 4; r cos A sin B = 3; r sin A = 5
34. Determine the area of a circle inscribed in a regular hexagon having an area of 240 square
The value of r is nearest to:
cm.
Ans. 7.07
Ans. 217.7
11. In a spherical triangle, angle B = 81°50’ and angle C = 94°30’. If side c = 90°, what is the
35. One side of a rectangle, inscribed in a circle of diameter 17 cm, is 8 cm. Find the area of the
value of side a?
rectangle.
Ans. 56°44’
Ans. 120 cm²
12. Given the sides of a triangle ABC: a = 36.3 cm, b = 23.9 cm, c = ?. The angle opposite side
36. Given a triangle ABC with side AB = 25cm, BC = 32 cm, and AC = 47 cm. Find the distance
a is 77.3°. Compute the value of side c in cm.
of the point of intersection of perpendicular bisectors to side BC.
Ans. 33.08
Ans. 25.4
13. The three sides of a triangle measures 36 cm, 18 cm, and 24 cm. What is the length of the
37. Three circles of radii 110, 140, and 220 are tangent to one another. What is the area of the
median drawn from the longest to opposite vertex?
triangle formed by joining the centers of the circles?
Ans. 11.22 cm
Ans. 39,904
14. If coversed sin Θ is 0.256855 then Θ is:
Ans. 48° 1
38. Solve for x if Arc tan (1 – x) + Arc tan (1 + x) = Arc tan 8 .
15. A car travels from point A northward for 30 minutes then eastward for one hour, then shifted
N 30° E. If the constant speed is 60 kph, how far directly from A, in km will be it after 2 Ans. 4
hours? 39. Triangle ABC is inscribed in a circle with side a = 60 cm, and angle BAC = 20° and angle
Ans. 93.6 ABC = 40°. Find the diameter of the circle.
16. The hypotenuse of a right triangle is 34 in. Find the lengths of the two legs if one leg is 14 in Ans. 175.4
longer than the other. 40. The lateral area of a regular pyramid with a square base is 1500 cm² and altitude of 20 cm.
Ans. 16 and 30 in Compute the base of the square.
17. The hypotenuse of a right triangle is 41 ft long and the area of the triangle is 180 ft². Find Ans. 30
the lengths of the two legs. 41. If x + y = 90°, what is the value of (sin x tan y)/ sin y tan x? ?
Ans. 9 and 40 feet Ans. 1/tan x
18. The three sides of a triangle measures 36 cm, 18 cm, and 24 cm. What is the length of the 42. Two chords AB and AC are equal and OB is also equal to OC where O is the center of the
median drawn from the shortest side to opposite vertex? circle circumscribing ABC. If the angle BOC is 228°, find the value of angle ABO.
Ans. 29.24 cm Ans. 33
19. A lighthouse is 10 units of length northwest of a dock. A ship leaves the dock at 8:00 am and 43. Simplify  1 - 1  (1 + Cos ) .
 Sin  tan  
travels west at 12 units of length per hour. At what time will the ship be 8 units of length from
the lighthouse? Ans. Sin 
Ans. 8:54 am 44. OAB is a sector of radius 24 units and has a central angle of 60°. A circle having a radius r
20. From the top of the building A the angle of elevation of the top of the building B is 46°. From is inscribed in the sector. Compute the radius of the circle.
the foot of the building B the angle of elevation of the top of building A is 28°. Both buildings Ans. 8 cm
are on level ground. If the height of the building B is 150 m, how far apart are the buildings 45. Determine the area of a spherical triangle ABC if A = 140°, B = 75°, C = 86° if it has a
in m? radius of 40 km.
Ans. 95.7 Ans. 3379 km²
21. An observer on the top of a cliff 150 m high above the angle of depression of two ships, 46. A wheel that is drawn by a belt is making 1 revolution per second (r/s). If the wheel is 18 cm
which are due north him, to be 20° 12’ and 47° 39’. Find the distance between the ships in in diameter, what is the linear velocity of the belt in cm/s?
meters. Ans. 18π or 57 cm/s
Ans. 270.96 47. The minute hand of a clock is 12 cm long. How far does the tip of the hand move during 20
22. From the third-floor window of the building, the angle of depression of an object on the min?
ground is 35° 58’, while from a sixth-floor window, 9.75 m above the first point of observation Ans. 25.1 cm
the angle of depression is 58° 35’. How far is the object from the building? 48. A central angle of a circle of radius 30 cm intercepts an arc of 6 cm. Express the central
Ans. 10.7 m angle θ in radians and in degrees.
23. At a certain point on the ground, the tower at the top of 20-m high building subtends an angle Ans. 11.46°
of 45°. At another point on the ground 25 m closer the building, the tower subtends an angle 49. A railroad curve is to be laid out on a circle. What radius should be used if the track is to
of 45°. Find the height of the tower. change direction by 25° in a distance of 120 m?
Ans. 101.85 m Ans. 275 m
24. A wooden flagpole is imbedded 3 m deep at corner A of a concrete horizontal slab ABCD, 50. A train is moving at the rate of 8 mi/h along a piece of circular track of radius 2500 ft. Through
square in form and measuring 20 ft on a side. A storm broke the flagpole at a point one what angle does it turn in 1 min?
meter above the slab and inclined toward corner C in the direction of the diagonal C. The Ans. 16.13°
vertical angles observed at the center of the slab and at corner C to the tip of the flagpole 51. Assuming the earth to be a sphere of radius 3960 mi, find the distance of a point 36°N
were 65° and 35°, respectively. What is the total length of the flagpole above the slab in latitude from the equator.
yards? Ans. 2488 mi
Ans. 5.61 52. Two cities 270 mi apart lie on the same meridian. Find their difference in latitude.
25. Two inaccessible objects A and B are each viewed from two stations, C and D, on the same Ans. 3°54.4’
side of AB and 562 m apart. The angle ACB is 62°12’, BCD is 41° 08’, ADB is 60°49’ and 53. A sector of a circle has a central angle of 50and an area of 605 cm². Find the radius of the
ADC is 34°51’. Find the required distance AB. circle.
Ans. 897.11 m Ans. 37.2 cm
26. It is 4.2 km from point A to the north end of the lake and 6.1 km from A to the south end of 54. A sector of a circle has a central angle of 80and a radius of 5 m. What is the area of the
the lake. The lake subtends an angle of 110° at A. The length of the lake from north to south sector?
is nearest to: Ans. 17.5 cm²
Ans. 8.5 km
TRIGONOMETRY
55. A wheel is turning at the rate of 48 r/min. Express this angular speed in (a) r/s, (b) rad/min, 78. In hilly or mountainous terrain, it is often difficult to determine heights. There are two
and (c) rad/s. problems: first, it may be impossible to find an expanse of level ground from which
Ans. (a) ⅘ r/s, (b) 301.6 rad/min (c) 5.03 rad/s measurements can be taken. Second, sighting may be limited. That is, points of reference
56. A wheel 4 ft in diameter is rotating at 80 r/min. Find the distance (in ft) traveled by a point on such as mountain peaks may be visible only from certain places. A geologist took
the rim in 1 s, that is, the linear velocity of the point (in ft/s). measurements at two points, B and C, 928 m apart. The angle of elevation from B to the top
Ans. 16.8 ft/s of the mountain at A is 47°10’. The angle of elevation from C to B is 8°40’, and the angle of
57. Find the diameter of a pulley which is driven at 360 r/min by a belt moving at 40 ft/s. elevation from C to A is 32°30’. It is known that the elevation at C is 1537 m. to the nearest
Ans. 2.12 ft meter, what is the elevation at A?
58. A point on the rim of a turbine wheel of diameter 10 ft moves with a linear speed of 45 ft/s. Ans. 2763 m
Find the rate at which the wheel turns (angular speed) in rad/s and in r/s. 79. Find the direction (in degrees) of the vectors (2, -3).
Ans. 1.43 r/s Ans. 303.7°
59. Determine the speed of the earth (in mi/s) in its course around the sun. Assume the earth’s 80. A pennant in a shape of an isosceles triangle is to be constructed for the Slidell High School
orbit to be a circle of radius 93,000,000 mi and 1 year = 365 days. Athletic Club and sold at a fund-raiser. The company manufacturing the pennant charges
Ans. 18.5 mi/s according to perimeter, and the athletic club has determined that a perimeter of 149 cm
60. In triangle ABC, tan A + tan B + tan C = 4, find the value of tan A tan B tan C. should make a nice profit. If each equal side of the triangle is twice the length of the third
Ans. 4 side, increased by 12 cm, find the lengths of the sides of the triangular pennant.
61. A hole 100 mm in diameter is to be punched out from a right circular cone having a diameter Ans. 25 cm, 62 cm, 62 cm
of 160 mm. Height of cone is 240 cm. Determine the length of the hole punched out. 81. Find the measures of the angles of a triangle if the measure of one angle is twice the
Ans. 90 mm measure of the second angle decreased by 12.
62. A cone is to be constructed from a section having a radius of 36 cm. and a central angle of Ans. The angles measure 64°, 32°, 84°.
210°. Find the radius of the cone. 82. Two frames are needed with the same perimeter: one frame in the shape of a square and
Ans. 21 cm one in the shape of an equilateral triangle. Each side of the triangle is 6 cm longer than each
63. An observer at point A observed the angle of elevation of 27° to the top of the pole. If he side of the square. Find the dimensions of each frame.
moves a horizontal distance x at B nearer to the pole, the angle of elevation of the pole at B Ans. The sides of the square are 18cm and the sides of the triangle are 24 cm.
is double than that of A. Find the height of the pole. 83. The official manual for the traffic signs is the Manual on Uniform Traffic Control Devices
Ans. x Sin 54° published by the Government Printing Office. The rectangular sign has a length of 12 in
64. ABC is a triangle having an angle A equal to 80°. Point D is a point inside the triangle. If BD more than twice its height. If the perimeter of the sign is 312 in, find its dimensions.
and CD are bisectors of angle B and C respectively, find the angle BDC. Ans. H = 48 in, L = 108 in
Ans. 130° 84. The measure of the largest angle of triangle is 80° more than the measure of the smallest
65. A man stands at C at a certain distance from a flagpole AB, which is 20 m. high. The angle angle, and the measure of the remaining angle is 10° more than the measure of the smallest
of elevation of the top of AB at C is 45°. The man then walks towards the pole at D. The angle. Find the measure of each angle.
angle of elevation of the top of the pole measured from D is 60°. Find the distance he had Ans. (30, 110, 40)
walked. 85. The perimeter of a triangle is 93 cm. If two sides are equally long and the third side is 9 cm
Ans. 8.46 longer than the others, find the lengths of the three sides.
66. A, B, and C are points on the same horizontal ground. PC is a vertical building of height 100 Ans. 28cm, 28cm and 37 cm
m. Angle ACB = 90°. The angle of elevations of P from A and B are 45° and 30° respectively. 86. One angle is three times its supplement increased by 20°. Find the measures of the two
D is a point along the line AB such that CD is perpendicular to AB. Determine the distance supplementary angles.
CD. Ans. 140° and 40°
Ans. 87 m. 87. Find an angle such that its supplement is equal to twice to its compliment increased by 50°.
67. A building and a tower stand 80 m. apart on a horizontal plane. AT a point midway between Ans. 50°
them, the angles of elevation of the top of the building and the tower are complimentary. If 88. The shorter leg of a right triangle is 3 cm less than the other leg. Find the length of the two
the tower is 60 m. high, what is the observed angle of elevation of the top of the building legs if the hypotenuse is 15 cm.
when observed at the foot of the tower? Ans. 12 cm and 9 cm
Ans. 18.46° 89. The hypotenuse of an isosceles right triangle is 2 cm longer than either of its legs. Find the
68. Ship Atlantis started sailing N.40°32’E at a rate of 3 mph. After 3 hours, Potomac started exact length of each side. (Hint: An isosceles right triangle is a right triangle whose legs are
from the same port going S.45°18’E at the rate of 4 mph. What would be the direction of the same length.)
Potomac from Atlantis 3 hours after Potomac start from the same port. Ans. a = 1, b = -4, c = -4
Ans. S.3°31’W 90. Find the values of cos θ and tan θ, given sin θ = 8/15 and in quadrant I.
69. Cebu Pacific plane flew from Busan, Korea whose latitude is 14°N and longitude of 121°30’E Ans. cos θ = 15/17 and tan θ = 8/15
on a course S.30°W and maintaining a uniform altitude. At what longitude will it cross the 91. Find the values of sin θ and tan θ, given cos θ = ⅚.
equator? Ans. sin θ = √11/6 and tan θ = √11/5
Ans. 113°33’E 92. Find the values of sin θ and cos θ, given tan θ = -¾.
70. What is the amplitude of the graph of y = cos x? Ans. for θ in Q II, sin θ = ⅗ and cos θ = -⅘ for θ in Q IV, sin θ = -⅗ and cos θ = ⅘
Ans. 2 93. Find sin θ, given cos θ = -⅘ and that tan θ is positive.
71. The area of a right triangle is 88.48 m² and its perimeter is 46.37 m. Find the sum of the Ans. sin θ = -3/5
shorter leg and the longer leg. 94. A support wire is anchored 12 m up from the base of a flagpole, and the wire makes a 15°
Ans. 27 angle with the ground. How long is the wire?
72. The Department of Energy reports that wind-produced electricity will jump to close to 1% of Ans. 46 m
the nation’s total electrical output by the year 2010, about ten times that produced today. 95. When the sun is 20° above the horizon, how long is the shadow cast by a building 50 m
“Wind farms” are springing up in many parts of the United States. A particular wind generator high?
can generate alternating current given by the equation I = 50 cos(120πt + 45π). Where t is Ans. 137 m
time in seconds and I is current in amperes. What is the current I (to two decimal places) 96. A tree 100 ft tall casts a shadow 120 ft long. Find the angle of elevation of the sun.
when t = 1.09 sec? (More will be said about alternating current in subsequent sections.) Ans. 40°
Ans. 40.45 amperes 97. A ladder leans against the side of a building with its foot 12 ft from the building. How far from
73. If a 6 cm shaft is rotating at 4,000 rpm (revolutions per minute), what is the speed of a particle the ground is the top of the ladder and how long is the ladder if it makes an angle of 70° with
on its surface (in cm per minute, to two significant digits)? the ground?
Ans. 75,000 cm/min Ans. 35 feet
74. A spotlight shining on a pond strikes the water so that the angle of incidence α is 23.5°. Find 98. From the top of a lighthouse 120 m above the sea, the angle of depression of a boat is 15°.
the refracted angle β. Use Snell’s law and the fact that n = 1.33 for water and n = 1.00 for How far is the boat from the lighthouse?
air. Ans. 448 m
Use: n2/n1 = sin α/sin β 99. Find the length of the chord of a circle of radius 20 cm subtended by a central angle of 150°.
Ans. 17.4° Ans. 39 cm
75. At what angle θ must a track be banked at a curve to eliminate sideways forces parallel to 100. Find the height of a tree if the angle of elevation of its top changes from 20to 40as the
the track, given that the curve has a radius of 525 m and the racing car is moving at 85.0 observer advances 75 ft toward its base.
m/sec (about 190 mph)? Ans. 48 ft
Ans. 54.5° 101. A tower standing on level ground is due north of point A and due west of point B, a distance
76. Express sin 40° cos 60° as a sum. c ft from A. If the angles of elevation of the top of the tower as measured from A and B are
Ans. ½ [sin 100° - sin 20°] α and β, respectively, find the height h of the tower.
77. The 25,000 lb Hubble space telescope was launched April 1990 and placed in a 380 mi Ans. ℎ = √(cot 2
𝑐

circular orbit above the earth’s surface. It completes one orbit every 97 min, going from a 𝛼) +(cot 𝛽)2

dawn-to-dusk cycle nearly 15 times a day. If the radius of the earth is 3,964 mi, what is the 102. If holes are to be spaced regularly on a circle, show that the distance d between the centers
linear velocity of the space telescope in miles per hour (mph)? of two successive holes is given by d = 2r sin (180/n), where r = the radius of the circle and
Ans. 17,000 mph n = the number of holes. Find d when r = 20 in and n =4.
Ans. 20√2
TRIGONOMETRY
103. Considering the earth as a sphere of radius 3960 mi, find the radius r of the 40th parallel of 125. A tower 125 ft high is on a cliff on the bank of a river. From the top of the tower, the angle of
latitude. depression of a point on the opposite shore is 28°40’, and from the base of the tower, the
Ans. 3030 mi angle of depression of the same point is 18°20’. Find the width of the river and the height of
104. Find the perimeter of a regular octagon inscribed in a circle of radius 150 cm. the cliff.
Ans. 918 cm Ans. The river is 580 ft wide, and the cliff is 192 ft high
105. To find the width of a river, a surveyor set up his surveying equipment at C on one bank and 126. A pilot wishes a course 15°0’ against a wind of 25 mi/h from 160°30’. Find his required
sighted across to a point B on the opposite bank; then, turning through an angle of 90, he heading and the groundspeed when the airspeed is 175 mi/h.
laid off a distance CA = 225 m. Finally, setting the equipment at A, he measured ∠CAB as Ans. 140°50’, 195 mi
48°20’. Find the width of the river. 127. Two forces of 17.5 and 22.5 lb act on a body. If their directions make an angle of 50°10’ with
Ans. 253 m each other, find the magnitude of their resultant and the angle that it makes with the larger
106. The line AD crosses a swamp. In order to locate a point on this line, a surveyor turned force.
through an angle 51°16’ at A and measured 1585 feet to a point C. He then turned through Ans. 36.3 lb, 21°40’
an angle of 90° at C and ran a line CB. If B is on AD, how far must he measure from C to 128. From A a pilot flies 125 km in the direction N38°20’W and turns back. Through an error, the
reach B? pilot then flies 125 km in the direction S51°40’E. How far and in what direction must the pilot
Ans. 1976 ft now fly to reach the intended destination A?
107. From a point A on level ground, the angles of elevation of the top D and bottom B of a Ans. the pilot must fly a course S45°20’W for 29.0 km in going from C to A.
flagpole situated on the top of a hill are measured as 47°54’ and 39°45’. Find the height of 129. The distances of a point C from two points A and B, which cannot be measured directly, are
the hill if the height of the flagpole is 115.5 ft. required. The line CA is continued through A for a distance 175 m to D, the line CB is
Ans. 349.3 ft continued through B for 225 m to E, and the distances AB = 300 m, DB = 326 m, and DE =
108. From the top of a lighthouse, 175 ft above the water, the angle of depression of a boat due 488 m are measured. Find AC and BC.
south is 18°50’. Calculate the speed of the boat if, after it moves due west for 2 min, the Ans. AC = 145 m and BC = 350 m
angle of depression is 14°20’. 130. Two adjacent sides of a parallelogram are 3473 and 4822 ft, and the angle between them is
Ans. 227 ft/min\ 72.23. Find the length of the longer diagonal.
109. A 500-lb barrel rests on an 11.2° inclined plane. What is the minimum force (ignoring friction) Ans. 6748 ft
needed to keep the barrel from rolling down the incline and what is the force the barrel exerts 131. Find the area of triangle ABC, given c = 23 cm, A = 20°, and C = 15°.
against the surface of the inclined plane? Ans. 200 cm²
Ans. 97.1 lb 132. Find the area of triangle ABC, given c = 23 cm, A = 20°, and B = 15°.
110. A motorboat moves in the direction N40°E for 3 h at 20 mi/h. How far north and how far east Ans. 41 cm²
does it travel? 133. Find the area of triangle ABC, given a = 112 m, b = 219 m, and A = 20°.
Ans. The boat travels 46 mi north and 39 mi east Ans. 10,800 m² or 4600 m²
111. Three ships are situated as follows: A is 225 mi due north of C, and B is 375 mi due east of 134. Find the area of triangle ABC, given A = 41°50’, a = 123 ft, and b = 96.2 ft.
C. What is the bearing of (a) of B from A and (b) of A from B? Ans. 5660 ft²
Ans. (a) S59°0’E (b) N59°0’W 135. Find the area of triangle ABC, given b = 27 yd, c = 14 yd, and A = 43°.
112. Three ships are situated as follows: A is 225 miles west of C while B, due south of C, bears Ans. 130 yd²
S25°10’E from A. (a) How far is B from A? (b) How far is B from C? (c) What is the bearing 136. Find the area of triangle ABC, given a = 14.27 cm, c = 17.23 cm, and B = 86°14’.
of A from B? Ans. 122.7 cm²
Ans. (a) 529 miles (b) 479 miles (c) N25°10’W 137. Find the area of triangle ABC, given a = 5.00 m, b = 7.00 m, and c = 10.0 m.
113. From a boat sailing due north at 16.5 km/h, a wrecked ship K and an observation tower T Ans. 16.2 m²
are observed in a line due east. One hour later the wrecked ship and the tower have bearings 138. Find the area of triangle ABC, given a = 1.017 cm, b = 2.032 cm, and c = 2.055 cm.
S34°40’E and S65°10’E. Find the distance between the wrecked ship and the tower. Ans. 1.006 cm²
Ans. 24.2 km 139. Find the area of an isosceles triangle with a base of 19.2 in and base angles of 23°10’ each.
114. A ship is sailing due east when a light is observed bearing N62°10’E. After the ship has Ans. 39.4 in²
traveled 2250 m, the light bears N48°25’E. If the course is continued, how close will the ship 140. In a quadrangular field ABCD, AB runs N62°10’E 11.4 m, BC runs N22°20’W 19.8 m, and
approach the light? CD runs S40°40’W 15.3 m. DA runs S32°10’E but cannot be measured. Find (a) the length
Ans. 2934 m of DA and (b) the area of the field.
115. A body at O is being acted upon by two forces, one of 150 lb due north and the other of 200 Ans. (a) 13.9 m (b) 214 m²
lb due east. Find the magnitude and direction of the resultant. 141. Three circles with centers A, B, and C have respective radii 50, 30, and 20 in and are tangent
Ans. The magnitude of the resultant force is 250 lb and its direction is N53°10’E. to each other externally. Find the area of the curvilinear triangle formed by the three circles.
116. An airplane is moving horizontally at 240 mi/h when a bullet is shot with speed 2750 ft/s at Ans. 142 in²
right angles to the path of the airplane. Find the resultant speed and direction of the bullet. 142. Find the area of the triangle ABC, given A = 37°10’, C = 62°30’, and b = 34.9.
Ans. the bullet travels at 2770 ft/s along a path making an angle of 82°40’ or 82.7° with the Ans. 331 square units
path of the airplane. 143. Find the area of the triangle ABC, given b = 28.6, c = 44.3, and B = 23.3°.
117. A river flows due south at 125 ft/min. A motorboat, moving at 475 ft/min in still water, is Ans. Two triangles are determined, their areas being 554 and 159 square units
headed due east across the river. (a) Find the direction in which the boat moves and its 144. Find the area of the triangle ABC, given a = 16.4, b =55.7, and C = 27.3°.
speed. (b) In what direction must the boat be headed in order to move due east, and what Ans. 209 square units
is its speed in that direction? 145. Find the area of the triangle ABC, given a = 255, b = 290, and c = 419.
Ans. (a) 491 ft/min, S75°20’E. (b) N74°40’E, 458 ft/min Ans. 36,400 square units
118. A man pulls a rope attached to a sled with a force of 100 lb. The rope makes an angle of 146. The centers of two circles with radii of 3 m and 5 m, respectively are 4 m apart. Find the
27° with the ground. (a) Find the effective pull tending to move the sled along the ground area of the portion of the smaller circle outside the larger circle.
and the effective pull tending to lift the sled vertically. (b) Find the force which the man must Ans. 10.05 m²
exert so that the effective force tending to move the sled along the ground is 100 lb. 147. Find the area in square centimetre of the largest square that can be cut from a sector of a
Ans. (a) Fh = 89 lb, Fv = 45 lb (b) 112 lb circle radius 8 cm and central angle 120°.
119. A block weighing W = 500 lb rests on a ramp inclined 29with the horizontal. (a) Find the Ans. 33.5
force tending to move the block down the ramp and the force of the block on the ramp. (b) 148. The perimeter of a small rectangular industrial lot is 140 m and its diagonal is 50 m. Find the
What minimum force must be applied to keep the block from sliding down the ramp? Neglect area of the lot in m²
friction. Ans. 1200
Ans. (a) F1 = 242 lb, F2 = 437 lb (b) 242 lb the ramp 149. In a spherical triangle ABC, A = 116°, B = 55° and C = 80°. Find the value of “a” in degrees.
120. The heading of an airplane is 75°, and the airspeed is 200 mi/h. Find the groundspeed and Ans. 114.83°
course if there is a wind of 40 mi/h from 165°. 150. BC is a chord 10 cm. long of a circle having AB as the diameter. CD is another chord with
Ans. 63°40’ angle BDC = 18°. What is the area of the circle?
121. The airspeed of an airplane is 200 km/h. There is a wind of 30 km/h from 270°. Find the Ans. 822.47
heading and groundspeed in order to track 0°. 151. A quadrilateral ABCD is inscribed in a circle with CD as the diameter. AB is parallel to CD
Ans. 351°20’ and AB is shorter than CD. If the value of the angle ABD is 40°, determine the value of the
122. There is a wind of 35 mi/h from 320°. Find the airspeed and heading in order that the angle ADB.
groundspeed and course be 250 mi/h and 50°, respectively. Ans. 10°
Ans. airspeed = 252 mi/hr, heading = 42° 152. A circle is inscribed in a triangle ABC and it touches AC at P. If AB = 14, AC = 10 and AP =
123. A and B are two points on opposite banks of a river. From A, a line AC= 275 m is laid off, 4, what is the length of the side BC?
and the angles CAB = 125°40’ and ACB = 48°50’ are measured. Find the length of AB. Ans. 16
Ans. 2160 mi 153. Determine the length of the angular bisector from vertex A to side BC of triangle ABC if c =
124. A ship is sailing due east when a light is observed bearing N62°10’E. After the ship has 14, a = 28 and b = 18 cm.
traveled 2250 m, the light bears N48°25’E. If the course is continued, how close will the ship Ans. 7.69
approach the light? 154. In triangle ABC, c = 15, a = 18, and b = 24. How far is the point of intersection of the angular
Ans. 2934 m bisectors from the side C.
Ans. 4.73
TRIGONOMETRY
155. In triangle ABC, c = 30, a = 36 and b = 48. The perpendicular bisectors of the sides intersect 180. Determine (a) amplitude, (b) period, and (c) phase shift, and sketch two cycles of y = -3
at point x. How far is x from side a? cos(2x - π) - 1
Ans. 15.92 Ans. (a) 3, (b) π, (c) π/2 to the right ½ unit upward
156. ABC is a circular sector with a central angle of 40° at its center at A and has a radius of 20 181. Find the frequency of the sine wave given by y = sin (524πx)
cm. BC is the arc. From the point B, a line is drawn to point D, the midpoint of AC. Find the Ans. 262
area of BCD in [Link]. 182. The angle of elevation of the top of a water tower from point A on the ground is 19.9°. From
Ans. 75.3 point B, 50.0 feet closer to the tower, the angle of elevation is 21.8°. What is the height of
157. Two sides of a parallelogram are 68 cm, and 83 cm. One of the diagonals is 42 cm long. the tower?
Determine the biggest interior angle of the parallelogram. Ans. 191 feet
Ans. 149.73° 183. In the late 1950s, the Soviets labored to develop a missile that could stop the U-2 spy plane.
158. Points A, B, C and D lies on the circle with a radius r. AD is a chord 5 cm long and the angle On May 1, 1960, Nikita S. Khrushchev announced to the world that the Soviets had shot
ACD = 15°, with AB as the diameter of the circle. Find the radius r. down Francis Gary Powers while Powers was photographing the Soviet Union from a U-2
Ans. 9.66 at an altitude of 14 miles. How wide a path on the earth’s surface could Powers see from
159. ABCD is a square of side 10 cm. Four congruent isosceles triangles are cut off from the four that altitude? (Use 3950 miles as the earth’s radius.)
corners so that the remaining portion is a regular octagon. Compute the equal sides of the Ans. 661.8 miles
isosceles triangle. 184. A 20-inch belt connects a pulley with a very small shaft on a motor. The distance between
Ans. 2.93 the shaft and the large pulley is 4 in. Assuming the shaft is a single point, find the radius of
160. Two tangents were drawn from T to a circle and has its point of tangency on the circle at A the large pulley to the nearest tenth of an inch
and B. The angle between the tangents is 54°. Point B is along the periphery of the circle Ans. 2.2 in
and is nearer to T than A and B. If the lines AB and BC are constructed, determine the angle 185. Find the exact value of sin (α + β), given that sin α = -3/5 and cos β = -1/3, with α in quadrant
between the line AB and BC at point B. IV and β in quadrant III.
Ans. 117° Ans. (3 - 8√2)/15
161. A corner lot of land is 122.5 m. on one street and 150 m. on the other street, the angle 186. A bush pilot left the Fairbanks Airport in a light plane and flew 100 miles toward Fort Yukon
between the two streets being 75°. The other two lines of the lot are respectively in still air on a course with a bearing of 18°. She then flew due east 1bearing 90!2 for some
perpendicular to the lines of the streets. What is the perimeter of the boundary of the lot? time to drop supplies to a snowbound family. After the drop, her course to return to Fairbanks
Ans. 481.6 m had a bearing of 225°. What was her maximum distance from Fairbanks?
162. A ferris wheel is 20 meters in diameter and makes one revolution every 4 minutes. For how Ans. 134.5 miles
many minutes of any revolution will your seat be above 15 meters? 187. A central angle 𝛼 in a circle of radius r intercepts a chord of length a. Write a formula for a
Ans. 4/3 minutes each revolution in terms of 𝛼 and r.
163. If a weight hanging on a string of length 3 feet swings through 5° on either side of the vertical,
Ans. a = 𝑟√2 − 2 cos 𝛼
how long is the arc through which the weight moves from one high point to the next high
188. Two draft horses are pulling on a tree stump with forces of 200 pounds and 300 pounds. If
point? [Hint: arc length = radius x angle spanned in radians]
the angle between the forces is 65°, then what is the magnitude of the resultant force? What
Ans. π/6 feet is the angle between the resultant and the 300-pound force?
164. A tree 50 feet tall casts a shadow 60 feet long. Find the angle of elevation Θ of the sun.
Ans.
Ans. 39.806°
189. The heading of an executive’s Lear jet has a bearing of 320°. The wind is 70 mph with a
165. A compact disk is 12 cm in diameter and rotates at 100 rpm (revolutions per minute) when
bearing of 190°. Given that the air speed of the plane is 400 mph, find the (a) drift angle, the
being played. The hole in the center is 1.5 cm in diameter. Find the speed in cm/min of a (b) ground speed, and the (c) course of the airplane.
point on the outer edge of the disk.
Ans. (a) 8.6° (b) 359.0 mph (c) 311.4°
Ans. 3770 cm/min
190. Use De Moivre’s theorem to simplify (-√3 + i)⁸. Write the answer in the form a + bi.
166. A weather satellite orbits the earth in a circular orbit 500 miles above the earth’s surface.
Ans. -128 + 128i√3
What is the radian measure of the angle (measured at the center of the earth) through which
191. One leg of a right triangle is 20 in and the hypotenuse is 10 in longer than the other leg. Find
the satellite moves in traveling 600 miles along its orbit? [Hint: the radius of the earth is 3960
the lengths of the unknown sides.
miles and s = r θ where θ is in rad.] Ans. 15 and 25 in
Ans. 0.1345 radians
192. An aviator flew 70 mi east from A to C. From C, he flew 100 mi north to B. Find the measure
167. Find two positive angles and two negative angles that are coterminal with -50°.
of the angle of the turn (to the nearest degree) that must be made at B to return to A.
Ans. The angles 310°,670°, -410°, and -770° are coterminal with -50°.
Ans. 145°
168. Find two positive and two negative angles that are coterminal with π/6. 193. A road is to be constructed so that it will rise 105 ft for each 1000 ft of horizontal distance.
Ans. 13π/6, 25π/6, -11π/6 and -23π/6
Find the measure of the angle of rise to the nearest degree, and the length of road to the
169. For four years, U-2 spy planes flew reconnaissance over the Soviet Union, collecting
nearest foot for each 1000 ft of horizontal distance.
information for the United States. The U-2, designed to fly at 70,000 feet—well out of range
Ans. 1006 ft
of Soviet guns—carried 12,000 feet of film in a camera that could photograph a path 2000
194. Sighting to the top of a building, Henry found the angle of elevation to measure 21°. The
miles long and about 600 miles wide. Find the central angle, to the nearest tenth of a degree,
ground is level. The transit is 5 ft above the ground and 200 ft from the building. Find the
that intercepts an arc of 600 miles on the surface of the earth (radius 3950 miles).
height of the building to the nearest foot.
Ans. 8.7°
Ans. 82 feet
170. What are the angular velocity in radians per second and the linear velocity in miles per hour 195. If the angle of elevation of the sun at a certain time measures 42°, find to the nearest foot
of the tip of a 22-inch lawnmower blade that is rotating at 2500 revolutions per minute?
the height of a tree whose shadow is 25 ft long.
Ans. 261.799 rad/sec, 163.624 mi/hr
Ans. 23 ft
171. What is the linear velocity in miles per hour of a point on the equator? 196. Standing at the top of a lighthouse 200 ft high, a lighthouse keeper sighted both an airplane
Ans. 1034 mi/hr.
and a ship directly beneath the plane. The angle of elevation of the plane measured 25°;
172. Find cos α, given that sin α = 3/5 and α is an angle in quadrant II.
the angle of depression of the ship measured 32°. Find (a) the distance d of the boat from
Ans. cos α = -⅘ the foot of the lighthouse, to the nearest 10 ft; (b) the height of the plane above the water, to
173. A weight on a certain spring is set in motion with an upward velocity of 3 cm per second from the nearest 10 ft.
a position 2 cm below equilibrium. Assume that for this spring and weight combination the Ans. (a) 320 ft (b) 350 ft
constant ω has a value of 1. (a) Write a formula that gives the location of the weight in cm 197. An observer on the top of a hill 250 ft above the level of a lake sighted two boats directly in
as a function of the time t in seconds, and (b) find the location of the weight 2 seconds after line. Find, to the nearest foot, the distance between the boats if the angles of depression
the weight is set in motion. noted by the observer measured 11° and 16°.
Ans. (a) x = -3 sin t + 2 cos t (b) 3.6 cm above the equilibrium position Ans. 141 ft
174. Find the amplitude of the functions y = sin x and y = 2 sin x. 198. Find the area of an equilateral triangle whose perimeter is 24.
Ans. 1 and 2 Ans. 16√3
175. Graph two cycles of y = sin (x + π/6) and determine the phase shift of the graph. 199. Find the ratio of the areas of two similar triangles (a) if the ratio of the lengths of two
Ans. phase shift = π/6 to the left corresponding sides is 3:5; (b) if their perimeters are 12 and 7. Find the ratio of the lengths
176. Graph two cycles of y = cos (x - π/4) + 2, and determine the phase shift of the graph. of a pair of (c) corresponding sides if the ratio of the areas is 4:9; (d) corresponding medians
Ans. π/4 to the right and two units upward if the areas are 250 and 10.
177. Graph two cycles of y = sin (2x) and determine the period of the function. Ans. (a) 9/25 (b) 144/49 (c) ⅔ (d) 5
Ans. π 200. In an equilateral triangle, (a) find the lengths of the radius, apothem, and side if the altitude
𝜋
178. Determine the period of y = cos (2 x). has length 6; (b) find the lengths of the side, apothem, and altitude if the radius is 9.
Ans. 4 Ans. (a) 4√3 (b) 13½
179. Determine (a) amplitude, (b) period, and (c) phase shift, and sketch two cycles of y = 2 sin
(3x + π) + 1.
Ans. (a) 2 (b) 2π/3 (c) -π/3