399 3.5 Rational Functions: Graphs, Applications, and Models 114. Waiting in Line Queuing theory (or waiting-line theory) investigates the problem of providing adequate service economically to customers waiting in line. Suppose customers arrive at a fast-food service window at the rate of 9 people per hour. With reasonable assumptions, the average time (in hours) that a customer will wait in line before being served is modeled by the rational function ƒ1x2 = 9 x1x - 92 , where x is the average number of people served per hour. A graph of ƒ1x2 for x 79 is shown in the figure. (a) Why is the function meaningless if the average number of people served per hour is less than 9? Suppose the average time to serve a customer is 5 min. (b) How many customers can be served in an hour? (c) How many minutes will a customer have to wait in line (on the average)? (d) Suppose we want to halve the average waiting time to 7.5 min A1 8 hrB. How fast must an employee work to serve a customer (on the average)? (Hint: Let ƒ1x2 = 1 8 and solve the equation for x. Convert the answer to minutes and round to the nearest hundredth.) How might this reduction in serving time be accomplished? 115. Braking Distance Braking distance for automobiles traveling at x miles per hour, where 20 … x … 70, can be modeled by the rational function d1x2 = 8710x2 - 69,400x + 470,000 1.08x2 - 324x + 82,200 . (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, 2nd ed., John Wiley & Sons.) (a) Use graphing to estimate x to the nearest unit when d1x2 = 300. (b) Complete the table for each value of x. 0 1 0.5 x y x = 9 People served Hours Average Waiting Time 10 20 x d1x2 x d1x2 20 50 25 55 30 60 35 65 40 70 45 (c) If a car doubles its speed, does the braking distance double or more than double? Explain. (d) Suppose that the automobile braking distance doubled whenever the speed doubled. What type of relationship would exist between the braking distance and the speed?
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