Question

The average number of vehicles waiting in line to enter a parking lot can be modeled by the function f (x )equalsStartFraction x squared Over 2 (1 minus x )EndFraction ​, where x is a number between 0 and 1 known as the traffic intensity. Find the rate of change of the number of vehicles waiting with respect to the traffic intensity for the intensities ​(a) xequals0.3 and ​(b) xequals0.6.

Answer 1

The rate of change of the number of vehicles waiting with respect to the traffic intensity for the intensities are:

a) 0.52

b) 2.63

Explanation:

The rate of change of a function at a given point P can be obtained by evaluating the 1st derivative of the function in P. Thus,

`f(x)=(x^2)/(2(1-x))`

`(df(x))/(dx) = (d((x^2)/(2(1-x))))/(dx) \\(df(x))/(dx)  =(x)/(1-x)  + (x^2)/(2(1-x)^2) \\(df(x))/(dx) = -(x(x-2))/(2(x-1)^2)`

Now for we just need to evaluate in each of the given points

a)
`(df(0.3))/(dx) =-(0.3(0.3-2))/(2(0.3-1)^2)\\\boxed{(df(0.3))/(dx) =0.520408 \approx 0.52}`

b)
`(df(0.6))/(dx) =-(0.6(0.6-2))/(2(0.6-1)^2)\\\boxed{(df(0.6))/(dx) = 2.625\approx 2.63}`