Precalculus word problem

Question

Suppose the average number of vehicles arriving at the main gate of an amusement park is equal to 10 per min, while the average number of vehicles 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 5 vehicles per minute, how many attendants are needed to keep the average wait to 15 seconds or less?

Answer 1

Part A:

We are basically given f(x) and asked to solve for x:

f(x)=15 seconds = 0.25 minutes = (x-5)/(x^2-10x)

Multiply both sides by that denominator:

.25x^2-2.5x=x-5

Combine like terms:

.25x^2-3.5x+5=0

Use the quadratic equation to solve for x

x=[-(-3.5)+- sqrt((-3.5^2)-4(.25)(5))]/2(.25)

x= 1.61 or x = 12.39

This equation is only good for x>10, so x = 12.39 cars through the gate per minute

Part B:

We just solved for where f(x) = .25 minutes, and found that at least 12.39 cars need to be let through the gate per minute to keep f(x) at or below 0.25 minutes, so now, to reframe the question, how do we get at least 12.39 cars through the gate per minute? With attendants. Because each attendant can get 5 cars through the gate per minute, we need:

(12.39 cars/minute)/(5 cars/minute) = at least 2.47 attendants, so at least 3 attendants