A homeowner has 32 feet of fencing to build three sides of a rectangular chicken coup. One side of the chicken coup will be against the barn, so the 32 feet of fence will only be used for three sides. What are the ideal length and width of the chicken coup that would maximize its area?

The ideal length is 16 feet

The ideal width is 8 feet

Step-by-step explanation:

The given parameters are;

The length of the fencing available to the homeowner = 32 feet

The number of sides of the rectangular chicken coup the fence will be applied = 3 sides

Let L represent the length and W, represent the width

Therefore, we have;

The perimeter of the fence = 2W + L = 32 feet

L = 32 - 2·W

The area is given by, A = Length, L × Width, W

A =   (32 - 2·W) × W = 32·W - 2·W²

Differentiating with respect to W and equating to 0 to find the maximum point gives;

dA/dW = d(32·W - 2·W²)/dW = 0

32 - 4·W = 0

4·W = 32

W = 32/4 = 8

W = 8 feet

Given that the sign of the variable in the equation of the derivative is negative, the value we get is the maximum point

The ideal width W = 8 feet

The length, L = 32 - 2·W = 32 - 2 × 8 = 16 feet

Therefore, the ideal length, L = 16 feet.