Question

A woman wants to build a rectangular garden. She plans to use a side of a shed for one side of the garden. She has 84 yd of fencing material.

What is the maximum area that will be enclosed?

Enter your answer in the box.
_____yd²

Answer 1

882 yd squared

Explanation:

Answer 2

Let x denote the length of the side of the garden which is covered fenced by a shed, and
`(A)/(x)` be the width of the garden.

The perimeter of a rectangle is given by 2(length + width)
i.e.
`2x + (A)/(x) = 84`
which gives:

`A = 84x - 2x^2`

For the area to be maximum, the differentiation of A with respect to x must be equal to 0.
i.e.
`(dA)/(dx) =84-4x=0 \\ 4x=84 \\ x=21`

Therefore, the maximum area of the garden enclosed is given by

`84(21)-2(21)^2=1764-2(441)=1764-882=882 \, yd^2`