Question

A farmer is constructing a rectangular pen with one additional fence across its width. Find the maximum area that can be enclosed with 360 yards of fencing

Answer 1

The maximum area of the pen is the highest area the pen can have

The maximum area is 16200 square yards

Let the dimension be x and y.

So, the perimeter is given as:

$\mathbf{P = 360}$

Because it has one additional fence, the perimeter is calculated as:

$\mathbf{2x + y = 360}$

Make y the subject

$\mathbf{y = 360 - 2x}$

The area is calculated as:

$\mathbf{A = xy}$

Substitute $\mathbf{y = 360 - 2x}$

$\mathbf{A = x(360 -2x)}$

Expand

$\mathbf{A = 360x -2x^2}$

Differentiate

$\mathbf{A' = 360 -4x}$

Set to 0

$\mathbf{360 -4x = 0}$

Rewrite as:

$\mathbf{4x = 360}$

Divide both sides by 4

$\mathbf{x = 90}$

Substitute 90 for x in $\mathbf{A = 360x -2x^2}$

$\mathbf{A = 360 \times 90 - 2 \times 90^2}$

$\mathbf{A = 16200}$

Hence, the maximum area is 16200 square yards

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