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:
[tex]\mathbf{P = 360}[/tex]
Because it has one additional fence, the perimeter is calculated as:
[tex]\mathbf{2x + y = 360}[/tex]
Make y the subject
[tex]\mathbf{y = 360 - 2x}[/tex]
The area is calculated as:
[tex]\mathbf{A = xy}[/tex]
Substitute [tex]\mathbf{y = 360 - 2x}[/tex]
[tex]\mathbf{A = x(360 -2x)}[/tex]
Expand
[tex]\mathbf{A = 360x -2x^2}[/tex]
Differentiate
[tex]\mathbf{A' = 360 -4x}[/tex]
Set to 0
[tex]\mathbf{360 -4x = 0}[/tex]
Rewrite as:
[tex]\mathbf{4x = 360}[/tex]
Divide both sides by 4
[tex]\mathbf{x = 90}[/tex]
Substitute 90 for x in [tex]\mathbf{A = 360x -2x^2}[/tex]
[tex]\mathbf{A = 360 \times 90 - 2 \times 90^2}[/tex]
[tex]\mathbf{A = 16200}[/tex]
Hence, the maximum area is 16200 square yards
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