SECTION 4.6 The Graph of a Rational Function 257 58. Drug Concentration The concentration C of a certain drug in a patient’s bloodstream t minutes after injection is given by ( ) = + C t t t 50 25 2 (a) Find the horizontal asymptote of ( ) C t . What happens to the concentration of the drug as t increases? (b) Using a graphing utility, graph ( ) = C C t . (c) Determine the time at which the concentration is highest. 59. Minimum Cost A rectangular area adjacent to a river is to be fenced in; no fence is needed on the river side. The enclosed area is to be 1000 square feet. Fencing for the side parallel to the river is $5 per linear foot, and fencing for the other two sides is $8 per linear foot; the four corner posts are $25 apiece. Let x be the length of one of the sides perpendicular to the river. (a) Write a function ( ) C x that describes the cost of the project. (b) What is the domain of C? (c) Use a graphing utility to graph ( ) = C C x . (d) Find the dimensions of the cheapest enclosure. 60. Doppler Effect The Doppler effect (named after Christian Doppler) is the change in the pitch (frequency) of the sound from a source ( )s as heard by an observer ( )o when one or both are in motion. If we assume both the source and the observer are moving in the same direction, the relationship is ′ = − − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ f f v v v v a o s where ′ = f perceived pitch by the observer = f actual a pitch of the source = v speed of sound in air (assume 772.4 mph) = v speed o of the observer = v speed s of the source Suppose that you are traveling down a road at 45 mph and you hear an ambulance (with siren) coming toward you from the rear. The actual pitch of the siren is 600 hertz (Hz). (a) Write a function ( ) ′ f vs that describes this scenario. (b) If ′ = f 620 Hz, find the speed of the ambulance. (c) Use a graphing utility to graph the function. (d) Verify your answer from part (b). Source: www.acs.psu.edu/drussell/ 61. Minimizing Surface Area United Parcel Service has contracted you to design a closed box with a square base that has a volume of 10,000 cubic inches. See the figure. (a) Express the surface area S of the box as a function of x. (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area? 62. Minimizing Surface Area United Parcel Service has contracted you to design an open box with a square base that has a volume of 5000 cubic inches. See the figure. (a) Express the surface area S of the box as a function of x. (b) Using a graphing utility, graph the function found in part (a). (c) What is the minimum amount of cardboard that can be used to construct the box? (d) What are the dimensions of the box that minimize the surface area? (e) Why might UPS be interested in designing a box that minimizes the surface area? 63. Cost of a Can A can in the shape of a right circular cylinder is required to have a volume of 500 cubic centimeters. The top and bottom are made of material that costs 6¢ per square centimeter, while the sides are made of material that costs 4¢ per square centimeter. (a) Express the total cost C of the material as a function of the radius r of the cylinder. (Refer to Figure 55.) (b) Graph ( ) = C C r . For what value of r is the cost C a minimum? 64. Material Needed to Make a Drum A steel drum in the shape of a right circular cylinder is required to have a volume of 100 cubic feet. (a) Express the amount A of material required to make the drum as a function of the radius r of the cylinder. (b) How much material is required if the drum’s radius is 3 feet? (c) How much material is required if the drum’s radius is 4 feet? (d) How much material is required if the drum’s radius is 5 feet? (e) Graph ( ) = A A r . For what value of r is A smallest? 65. Tennis Anyone? To win a game in tennis, a player must win four points. If both players have won three points, the play continues until a player is ahead by two points to win the game. The model ( ) ( ) = − + − + − + P x x x x x x x 8 28 34 15 2 2 1 4 3 2 2 represents the probability P of a player winning a game in which the player is serving the game and x is the probability of winning a point on serve. The player serving is the first to put the ball in play. Source: Chris Gray, “Game, set and stats,” Significance, February 2015. (a) What is the probability that a player who is serving will win the game if the probability of the player winning a point on serve is 0.64? (b) Find and interpret ( ) P 0.62 . (c) Solve ( ) = P x 0.9. (d) Graph ( ) = P P x for ≤ ≤ x 0 1. Describe what happens to P as x approaches 1. x x y x x y Credit: Perry Mastrovito/Getty Images
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