Suppose the average number of vehicles arriving at the main gate of an amusement

Question

Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per minute, while the average number of vehicles being admitted through the gate per minute is equal to x. Then the average waiting time in minutes for each vehicle at the gate is given by f(x) = x - 5/x^2 - 10x, where x > 10. (a) Estimate the admittance rate x that results in an average wait of 15 seconds. (b) If one attendant can serve 3 vehicles per minute, how many attendants are needed to keep the average wait time 15 seconds or less? (a) Estimate the admittance rate x that results in an average The admittance is about vehicles per minute. (Round to the nearest tenth as needed.) (b) if one attendant can serve 3 vehicles per minute, how many attendants are needed to keep the average wait time 15 seconds or less? The amusement park needs at least attendants. (Round up to the nearest whole number as needed)

Explanation / Answer

f(x) = (x - 5) / (x^2 -10x)
f(x) = 15 seconds = 0.25 minutes
So: 0.25 = (x-5) / (x^2 - 10x)
0.25x^2 - 2.5x = x - 5
x^2 - 10x = 4x - 20
x^2 - 14x + 20 = 0
Using the quadratic formula to find x:
x = [14 +/- sqrt(196 - 4(1)(20))]/(2(1))
x = 1.61, 12.39
So the answer can be either 1.61 cars per minute or 12.39 cars per minute.
Since x must be greater than 10, the answer must be 12.39 cars per minute.

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