There is 326 ft of fences

We need to build a den for the goats. It needs to be as big as possible. (Most Square feet)

A) what is the max area the goats can have using 326 ft of fences?

B) what dimensions give us that area?

If the den is in rectangular shape, the perimeter is twice the sum of the length and the width.

P = 2L + 2W

Substituting,

326 = 2L + 2W; 163 = L + W; W = 163 - L

Then, the area is equal to,

A = L x W = L x (163 - L)

A = 163L - L²

Getting the derivative of the area and equating the derivative to zero,

dA = 163 - 2L = 0; L = 81.5 ft

Thus, the dimensions of the den is 81.5 ft x 81.5 ft

The maximum area is equal to 6642.25 ft².