Question

Palo alto college is planning to construct a rectangular parking lot on land bordered on one side by a highway. the plan is to use 720 feet of fencing to fence off the other three sides. what dimensions should the lot have if the enclosed area is to be a​ maximum?

Answer 1

The length of the 3 sides has a total dimension of 720 ft. One dimension, the length l, only has one side enclosed. The other dimension, the width w, has 2 sides enclosed. So,

720 ft = l + 2w

Rearranging in terms of l:

l = 720 - 2w

Then the area equals length times width, or:

A = (720-2w)(w) = 720w - 2w^2

To get the maximum area, we take the derivative of the Area equation and set the derivative equal to 0: dA/dw = 0

dA/dw = 720 - 4w = 0

720 - 4w = 0
4w = 720

w = 180 ft

Calculating for l:

l = 720 – 2w

l = 720 – 2(180)

l = 360 ft

Therefore to get the maximum enclosed area, the width (2 sides) should be 180 ft while the length (1 side) is 360 ft.