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Michael Sullivan, Brief Calculus: An Applied Approach 8/e, Wiley, 2004

Section 0.7

33. In the figure shown, ABCD is a square, with each side of length 6 feet. The width of the border (shaded portion) between the outer square EFGH and ABCD is 2 feet. Find the area of the border.

34. Refer to the figure above Problem 35. Square ABCD has an area of 100 square feet; square BEFG has an area of 16 square feet. What is the area of the triangle CGF?

35. A Norman window consists of a rectangle surmounted by a semicircle. Find the area of the Norman window shown in the illustration. How much wood frame is needed to enclose the window?

36. A circular swimming pool, 20 feet in diameter, is enclosed by a wooden deck that is 3 feet wide. What is the area of the deck? How much fence is required to enclose the deck?

37. The conning tower of the U.S.S. Silversides, a World War II submarine now permanently stationed in Muskegon, Michigan, is approximately 20 feet above sea level. How far can you see from the conning tower?

38. A person who is 6 feet tall is standing on the beach in Fort Lauderdale, Florida, and looks out onto the Atlantic Ocean. Suddenly, a ship appears on the horizon. How far is the ship from shore?

Section 0.8

33. A major league baseball “diamond” is actually a square, 90 feet on a side (see the figure). What is the distance directly from home plate to second base (the diagonal of the square)?

34. The layout of a Little League playing field is a square, 60 feet on a side. How far is it directly from home plate to second base (the diagonal of the square)?

35. Refer to Problem 33. Overlay a rectangular coordinate system on a major league baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base.
(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.
(b) If the right fielder is located at (310, 15), how far is it from the right fielder to second base?
(c) If the center fielder is located at (300, 300), how far is it from the center fielder to third base?

36. Refer to Problem 34. Overlay a rectangular coordinate system on a Little League baseball diamond so that the origin is at home plate, the positive x-axis lies in the direction from home plate to first base, and the positive y-axis lies in the direction from home plate to third base.
(a) What are the coordinates of first base, second base, and third base? Use feet as the unit of measurement.
(b) If the right fielder is located at (180, 20), how far is it from the right fielder to second base?
(c) If the center fielder is located at (220, 220), how far is it from the center fielder to third base?

37. A Dodge Intrepid and a Mack truck leave an intersection at the same time. The Intrepid heads east at an average speed of 30 miles per hour, while the truck heads south at an average speed of 40 miles per hour. Find an expression for their distance apart d (in miles) at the end of t hours.

38. A hot-air balloon, headed due east at an average speed of 15 miles per hour and at a constant altitude of 100 feet, passes over an intersection (see the figure). Find an expression for the distance d (measured in feet) from the balloon to the intersection t seconds later.

69. According to the American Automobile Association (AAA), the average cost of operating a standard-sized car, including gasoline, oil, tires, and maintenance increased to $0.122 per mile in 2000. Write an equation that relates the average cost C of operating a standard-sized car and the number x of miles it is driven.

70. The cost of renting a truck is $280 per week plus a charge of $0.30 per mile driven. Write an equation that relates the cost C for a weekly rental in which the truck is driven x miles.

70. Commonwealth Edison Company supplies electricity to residential customers for a monthly customer charge of $7.58 plus 8.275 cents per kilowatt-hour for up to 400 kilowatt-hours.
(a) Write an equation that relates the monthly charge C, in dollars, to the number x of kilowatt-hours used in a month, 0
£ x £ 400.
(b) Graph this equation.
(c) What is the monthly charge for using 100 kilowatt hours?
(d) What is the monthly charge for using 300 kilowatt hours?

74. Dan receives $375 per week for selling new and used cars at a car dealership in Omaha, Nebraska. In addition, he receives 5% of the profit on any sales he generates. Write an equation that relates Dan’s weekly salary S when he has sales that generate a profit of x dollars.

75. The cost to the Chicago Tribune for Sunday home delivery is approximately $0.53 per newspaper with fixed costs of $1,070,000. Write an equation that relates the cost C and the number x of copies delivered.

77. The relationship between Celsius (°C) and Fahrenheit (°F) degrees for measuring temperature is linear. Find an equation relating °C and °F if 0°C corresponds to 32°F and 100°C corresponds to 212°F. Use the equation to find the Celsius measure of 68°F.

80. A cereal company finds that the number of people who will buy one of its products the first month it is introduced is linearly related to the amount of money it spends on advertising. If it spends $400,000 on advertising, then 100,000 boxes of cereal will be sold, and if it spends $600,000, then 140,000 boxes will be sold.
(a) Write an equation describing the relation between the amount spent on advertising and the number of boxes sold.
(b) How much advertising is needed to sell 200,000 boxes of cereal?

Section 1.4

37. Sprint PCS offers a monthly cellular phone plan for $39.99. It includes 350 anytime minutes plus $0.25 per minute for additional minutes. The following function is used to compute the monthly cost for a subscriber where x is the number of anytime minutes used. Compute the monthly cost of the cellular phone for the following anytime minutes:
(a) 200
(b) 365
(c) 351

38. First-class Letter According to the U.S. Postal Service, first-class mail is used for personal and business correspondence. Any mailable item may be sent as first-class mail. It includes postcards, letters, large envelopes, and small packages. The maximum weight is 13 ounces. The following function is used to compute the cost of mailing a first-class item. where x is the weight of the item in ounces. Compute the cost of mailing the following items first-class:
(a) A letter weighing 4.3 ounces
(b) A postcard weighing 0.4 ounces
(c) A package weighing 12.2 ounces

Section 2.1

37. Maximizing Revenue Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R (in dollars) is
 

What unit price should be established for the dryer to maximize revenue? What is the maximum revenue?

38. The John Deere company has found that the revenue from sales of heavy-duty tractors is a function of the unit price p that it charges. If the revenue R is
 


what unit price p should be charged to maximize revenue? What is the maximum revenue? <39> Demand Equation The price p and the quantity x sold of a certain product obey the demand equation
 

(a) Express the revenue R as a function of x. (Remember, R = xp.)
(b) What is the revenue if 200 units are sold?
(c) What quantity x maximizes revenue? What is the maximum revenue?
(d) What price should the company charge to maximize revenue?

43. David has available 400 yards of fencing and wishes to enclose a rectangular area.
(a) Express the area A of the rectangle as a function of the width x of the rectangle.
(b) For what value of x is the area largest?
(c) What is the maximum area?

45. A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed? (See the figure.)

46. A farmer with 2000 meters of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?

47. A projectile is fired from a cliff 200 feet above the water at an inclination of 45° to the horizontal, with a muzzle velocity of 50 feet per second. The height h of the projectile above the water is given by
 

where x is the horizontal distance of the projectile from the base of the cliff.
(a) How far from the base of the cliff is the height of the projectile a maximum?
(b) Find the maximum height of the projectile.
(c) How far from the base of the cliff will the projectile strike the water?

49. A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges 90°. What depth will provide maximum cross-sectional area, allowing the most water to flow?

50. A Norman window has the shape of a rectangle surmounted by a semicircle of diameter equal to the width of the rectangle (see the figure). If the perimeter of the window is 20 feet, what dimensions will admit the most light (maximize the area)?

51. A track and field playing area is in the shape of a rectangle with semicircles at each end (see the figure). The inside perimeter of the track is to be 400 meters. What should the dimensions of the rectangle be so that the area of the rectangle is a maximum?

52. A special window has the shape of a rectangle surmounted by an equilateral triangle (see the figure). If the perimeter of the window is 16 feet, what dimensions will admit the most light?

53. The function H(x) = -1.01x2 + 114.3x + 451.0 models the number of individuals who engage in hunting activities whose annual income is x thousand dollars.
(a) What is the income level for which there are the most hunters? Approximately how many hunters earn this amount?

63. Between 12:00 PM and 1:00 PM, cars arrive at Citibank’s drive-thru at the rate of 6 cars per hour (0.1 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 12:00 PM:
(a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM).
(b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM).
(c) What value does F approach as t becomes unbounded in the positive direction?

64. Between 5:00 PM and 6:00 PM, cars arrive at Jiffy Lube at the rate of 9 cars per hour (0.15 car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 5:00 PM:
(a) Determine the probability that a car will arrive within 15 minutes of 5:00 PM (that is, before 5:15 PM).
(b) Determine the probability that a car will arrive within 30 minutes of 5:00 PM (before 5:30 PM).
(c) What value does F approach as t becomes unbounded in the positive direction?

65. Between 5:00 PM and 6:00 PM, cars arrive at McDonald’s drive-thru at the rate of 20 cars per hour. The following formula from probability can be used to determine the probability that x cars will arrive between 5:00 PM and 6:00 PM.
(a) Determine the probability that x = 15 cars will arrive between 5:00 PM and 6:00 PM.
(b) Determine the probability that x = 20 cars will arrive between 5:00 PM and 6:00 PM.

Section 2.4

97. The formula D = 5e-0.4h can be used to find the number of milligrams D of a certain drug that is in a patient's bloodstream h hours after the drug has been administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections?

98. A model for the number of people N in a college community who have heard a certain rumor is
N=P(1 - e-0.15d)
where P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a community of 1000 students, how many days will elapse before 450 students have heard the rumor?

99. The equation governing the amount of current I (in amperes) after time t (in seconds) in a simple RL circuit consisting of a resistance R (in ohms), an inductance L (in henrys), and an electromotive force E (in volts) is
 

If E = 12 volts, R = 10 ohms, and L = 5 henrys, how long does it take to obtain a current of 0.5 ampere? Of 1.0 ampere?

100. Psychologists sometimes use the function L(t)=A(1 – e-kt) to measure the amount L learned at time t. The number A represents the amount to be learned, and the number k measures the rate of learning. Suppose that a student has an amount A of 200 vocabulary words to learn. A psychologist determines that the student learned 20 vocabulary words after 5 minutes.
(a) Determine the rate of learning k.
(b) Approximately how many words will the student have learned after 10 minutes?
(c) After 15 minutes?
(d) How long does it take for the student to learn 180 words?

101. U.S. Population According to the U.S. Census Bureau, the population of the United States is projected to be 298,710,000 on January 1, 2010. Suppose this projection is correct, but after January 1, 2010, the population grows according to P(t) = 298,710,000 + 10,000,000log t. Project the population, to the nearest thousand, on January 1, 2020. Hint: t = 1 corresponds to the year 2010.

102. A reasonable projection for the population of the United States on January 1, 2025, is 336,566,000.
(a) If 2025 is taken as year 1, and the formula for year 2025 onward that represents a new trend in the population growth is P(t) = 336,566,000 + 8,000,000log t, what would be the population in 2045?
(b) Is this higher or lower than the current U.S. Census Bureau's estimate of 363,077,000?

109. The concentration of alcohol in a person's blood is measurable. Suppose that the risk R (given as a percent) of having an accident while driving a car can be modeled by the equation R=3ekx where x is the variable concentration of alcohol in the blood and k is a constant.
(a) Suppose that a concentration of alcohol in the blood of 0.06 results in a 10% risk (R = 10) of an accident. Find the constant k in the equation.
(b) Using this value of k, what is the risk if the concentration is 0.17?
(c) Using the same value of k, what concentration of alcohol corresponds to a risk of 100%?
(d) If the law asserts that anyone with a risk of having an accident of 15% or more should not have driving privileges, at what concentration of alcohol in the blood should a driver be arrested and charged with a DUI?

Section 2.6

9. If $1000 is invested at 2% compounded continuously, what is the amount after 1 year? How much interest is earned?
10. If $2000 is invested at 5% compounded continuously, what is the amount after 5 years? How much interest is earned?

11. If a bank pays 3% compounded continuously, how much should be deposited now to have $5000
(a) 4 years later?
(b) 8 years later?

14. What annual rate of interest compounded continuously is required to double an investment in 10 years?

15. Approximately how long will it take to triple an investment at 9% compounded continuously?

17. What principal is needed now to get $1000 in 1 year at 9% compounded continuously? How much should be invested to get $1000 in 2 years?

18. Laura wishes to have $8000 available to buy a car in 3 years. How much should she invest in a savings account now so that she will have enough if the bank pays 8% interest compounded continuously?

19. Tami and Todd will need $40,000 for a down payment on a house in 4 years. How much should they invest in a savings account now so that they will be able to do this? The bank pays 3% compounded continuously.

20. A newborn child receives a $3000 gift toward a college education. How much will the $3000 be worth in 17 years if it is invested at 10% compounded continuously?

21. What annual rate of interest compounded continuously is required to triple an investment in 5 years?

Section 4.1

57. A dive bomber is flying from right to left along the graph of y = x2. When a rocket bomb is released, it follows a path that approximately follows the tangent line. Where should the pilot release the bomb if the target is at (1, 0)?

Section 4.4
30. The demand function for a certain calculator is given by  where x (in thousands of units) is the quantity demanded per week and d(x) is the unit price in dollars.
(a) Find d(x).
(b) Find d(10), d(15), and d(20) and interpret your results.
(c) Find the revenue function.
(d) Find the marginal revenue.
31. The price p in dollars per pound when x pounds of a certain commodity are demanded is  
Find:
(a) The rate of change of price with respect to x.
(b) The revenue function.
(c) The marginal revenue.
(d) The marginal revenue at x = 10 and at x = 40.

73. At the Super Bowl, the demand for game-day t-shirts is given by


where p is the price of the shirt in dollars and x is the number of shirts demanded.
(a)   At what price can 1000 t-shirts be sold?
(b)   At what price can 5000 t-shirts be sold?
(c)    Find the marginal demand for 1000 t-shirts and interpret the answer.
(d)   Find the marginal demand for 5000 t-shirts and interpret the answer.
(e)    Find the revenue function R = R(x).
(f)    Find the marginal revenue from selling 1000 t-shirts and interpret the answer.
(g)   Find the marginal revenue from selling 5000 t-shirts and interpret the answer.
(h)   If each t-shirt costs $4, find the profit function P = P(x).
(i)    What is the profit if 1000 t-shirts are sold?
(j)    What is the profit if 5000 t-shirts are sold?



Section 4.6

59. A large container is being filled with water. After t hours there are 8t − 4t1/2 liters of water in the container. At what rate is the water filling the container (in liters per hour) when t = 4?


60. An object is propelled vertically upward with an initial velocity of 39.2 meters per second. The distance s (in meters) of the object from the ground after t seconds is
s = s(t) = -4.9t2 + 39.2t

(a) What is the velocity of the object at any time t?
(b) When will the object reach its highest point?
(c) What is the maximum height?

(d) What is the acceleration of the object at any time t?
(e) How long is the object in the air?
(f) What is the velocity of the object upon impact?
(g) What is the total distance traveled by the object?

Section 5.2

35. The cumulative ticket sales for the 10 days preceding a popular concert is given by S(x) = 4x2 + 50x + 5000
where x represents the 10 days leading up to the concert and 1 ≤ x ≤ 10. Show that S is an increasing function.

36. At Dan’s Toy Store, the revenue R, in dollars, derived from selling x electric trucks is R(x) = –0.005x^2 + 20 x
(a)   Determine where the graph of R is increasing and where it is decreasing.
(b)   How many trucks need to be sold to maximize revenue?
(c)    What is the maximum revenue?
(d)   Graph the function R.

37. The weekly revenue R, in dollars, from selling x calculators is R(x) = –20x2 + 1000x
(a)   Determine where the graph of R is increasing and where it is decreasing.
(b)   How many calculators need to be sold to maximize revenue?
(c)    What is the maximum revenue?
(d)   Graph the function R.

42. Suppose the cost C of producing x items is given by the function .
(a)   Find the marginal cost function.
(b)   Show that the marginal cost is a decreasing function.
(c)    Find the average cost function  .

(d)   Show that the average cost function is a decreasing function.


Section 5.2

62. The cost function for producing x items is C(x) = x2 – 3x + 625
(a)   Find the average cost function.
(b)   What is the minimum average cost?
(c)    Find the marginal cost function.
(d)   Graph the average cost function and the marginal cost function on the same set of axes. Label their point of intersection.


Section 5.4

26. An open box with a square base is to be made from a square piece of cardboard 24 centimeters on a side by cutting out a square from each corner and turning up the sides. Find the dimensions of the box that yield the maximum volume.

30. A cylindrical container is to be produced that will have a capacity of 10 cubic meters. The top and bottom of the container are to be made of a material that costs $2 per
square meter, while the side of the container is to be made of material costing $1.50 per square meter. Find the dimensions that will minimize the total cost of the container.

31. A telephone company is asked to provide telephone service to a customer whose house is located 2 kilometers away from the road along which the telephone lines run. The nearest telephone box is located 5 kilometers down this road. If the cost to connect the telephone line is $50 per kilometer along the road and $60 per kilometer away from the  road, where along the road from the box should the company connect the telephone line so as to minimize construction cost?

32. A small island is 3 kilometers from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row her boat 2.5 kilometers per hour and can walk 4 kilometers per hour, where should she land her boat in order to arrive in the shortest time at a town 12 kilometers down the shore from P?

33. A truck has a top speed of 75 miles per hour and, when traveling at the rate of x miles per hour, consumes gasoline at the rate of   gallon per mile. If the length of the trip is 200 miles and the price of gasoline is $1.60 per gallon, the cost is


where C(x) is measured in dollars. What is the most economical speed for the truck? Use the interval [10, 75].

34. If the driver of the truck in Problem 33 is paid $8 per hour, what is the most economical speed for the truck?

35. A printer plans on having 50 square inches of printed matter per page and is required to allow for margins of 1 inch on each side and 2 inches on the top and bottom. What are the most economical dimensions for each page if the cost per page depends on the area of the page?

36. A window is to be made in the shape of a rectangle surmounted by a semicircle with diameter equal to the width of the rectangle. See the figure. If the perimeter of the window is 22 feet, what dimensions will let in the most light?

38. A truck has a top speed of 75 miles per hour and, when traveling at the rate of x miles per hour, consumes gasoline at the rate of  

gallon per mile. This truck is to be taken on a 200 mile trip by a driver who is to be paid at the rate of b dollars per hour plus a commission of c dollars. Since the time required for this trip at x miles per hour is 200/x , the total cost, if gasoline costs a dollars per gallon, is



Find the most economical possible speed under each of the following sets of conditions:
(a)   b = 0, c = 0
(b)   a = 1.50, b = 8.00, c = 500
(c)    a = 1.60, b = 10.00, c = 0

39. Prove that a cylindrical container of fixed volume V requires the least material (minimum surface area) when its height is twice its radius.

40. An airplane crosses the Atlantic Ocean (3000 miles) with an airspeed of 500 mph.
(a)   Find the time saved with a 25 mile per hour tail wind.
(b)   Find the time lost with a 50 mile per hour head wind.
(c)    If the cost per passenger is where x is


the ground speed and C(x) is the cost in dollars, what is the cost per passenger when there is no wind?
(d)   What is the cost with a tail wind of 25 miles per hour?
(e)    What is the cost with a head wind of 50 miles per hour?
(f)    What ground speed minimizes the cost?
(g)   What is the minimum cost per passenger?

41. The concentration C of a certain drug in the bloodstream t hours after injection into muscle tissue is given by


When is the concentration greatest?

 

 

 

 

 

 

 

 

 

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