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

Let x denote the length of the side of the garden which is covered fenced by a shed, and $\frac{A}{x}$ be the width of the garden.

The perimeter of a rectangle is given by 2(length + width)
i.e. $2x + \frac{A}{x} = 84$
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. $\frac{dA}{dx} =84-4x=0 \\ 4x=84 \\ x=21$

Therefore, the maximum area of the garden enclosed is given by
$84(21)-2(21)^2=1764-2(441)=1764-882=882 \, yd^2$