Question

Let the average number of vehicles arriving at the gate of an amusement park per minute be equal to​ k, and let the average number of vehicles admitted by the park attendants be equal to r.​ Then, the average waiting time T​ (in minutes) for each vehicle arriving at the park is given by the rational function defined by the equation
T(r)= (2r-K) ÷ (2r²- 2kr), where r>k


(a) It is known from experience that on Saturday afternoon k= 26. Estimate the admittance rate r that is necessary to keep the average waiting time T for each vehicle to 30 sec.

​(b) If one park attendant can serve 5.3 vehicles per​ minute, how many park attendants will be needed to keep the average wait to 30​ sec?

Answer 1

Step-by-step explanation:

The given rational function defined by the equation:

$T(r)=\frac{(2r-k)}{(2r^2- 2kr)}$

where :k = average number of vehicles arriving at the gate per minute

r = average number of vehicles admitted by the park attendants

T =  the average waiting time in minutes for each vehicle

a) k = 26 , r = ?, T = 30 seconds

T(r) = 30 seconds

$30=\frac{(2r-30)}{(2r^2- 2\times 30r)}$

$60r^2-1800r=2r-30$

$60r^2-1802r+30=0$

on solving:

r = 0.016657 , 30.017

r = 30.017 (accept, given that r > k )

Admittance rate r that is necessary to keep the average waiting time T for each vehicle to 30 sec is 30.017.

b) k = 5.3, r = ?, T = 30 seconds

T(r) = 30 seconds

$30=\frac{(2r-5.3)}{(2r^2- 2\times 5.3r)}$

$60r^2-318r=2r-5.3$

$60r^2-318r-2r+5.3=0$

$60r^2-320r+5.3=0$

r = 0.016657 , 5.31672

r = 5.31672 (accept, given that r > k )

5.31672 park attendants will be needed to keep the average wait to 30​ seconds.