Joey wants to build a rectangular garden. He plans to use a side of a river for one side of the garden, he will not place fencing along this side of the garden. He has 92 yards of fencing material.

What is the maximum area that will be enclosed?

We let x and y be the measures of the sides of the rectangular garden. The perimeter subtracted with the other side should be equal to

2x + y = 92

The value of y in terms of x is equal to,

y = 92 - 2x

The area is the product of the two sides,

A = xy

Substituting,

A = x (92 - 2x) = 92x - 2x2

Solving for the derivative and equating to zero,

0 = 92 - 4x; x = 23

Therefore, the area of the garden is,

A = 23 (92 - 2 (23)) = 1058 yard 2