Find a function that models the area A of the corral in terms of the width x of the corral.

Question

Carol has 1600 ft of fencing to fence in a rectangular horse corral.

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x \ \

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800-x

(a) Find a function that models the area A of the corral in terms of the width x of the corral.

A(x)=

(b) Find the dimensions of the rectangle that maximize the area of the corral.

Width-

Length-

Answer 1

2x + 2y = 1600 ft.

A = xy

Rearrange the first one to y = 800 - x

Substitute into A = xy to get A = x (800 - x) (which answers part a)

Rewrite as A = 800x - x².

This is a parabola, and there are many methods to determine the maximum.

I'll take the derivative and set it equal to zero:

dA/dx = 800 - 2x = 0

x = 400 ft.

Substitute back in to y = 800 - x and we get y = 400 ft.