The area of a shape is the amount of space it occupies.
The dimensions to build the largest possible athletic complex is 100 by 50 feet
Assume the length of the fence is y.
So, we have:
[tex]\mathbf{Area = xy}[/tex]
[tex]\mathbf{Perimeter = 2y + 4x}[/tex]
Substitute 400 for perimeter
[tex]\mathbf{2y + 4x = 400}[/tex]
Divide through by 2
[tex]\mathbf{y + 2x = 200}[/tex]
Make y the subject
[tex]\mathbf{y = 200 -2x}[/tex]
Substitute [tex]\mathbf{y = 200 -2x}[/tex] in [tex]\mathbf{Area = xy}[/tex]
[tex]\mathbf{Area =x(200 - 2x)}[/tex]
[tex]\mathbf{Area =200x - 2x^2}[/tex]
Differentiate
[tex]\mathbf{A' =200 - 4x}[/tex]
Set to 0
[tex]\mathbf{200 - 4x = 0}[/tex]
Collect like terms
[tex]\mathbf{4x = 200}[/tex]
Divide both sides by 4
[tex]\mathbf{x = 50}[/tex]
Recall that: [tex]\mathbf{y = 200 -2x}[/tex]
So, we have:
[tex]\mathbf{y = 200 - 2 \times 50}[/tex]
[tex]\mathbf{y = 200 - 100}[/tex]
[tex]\mathbf{y = 100}[/tex]
Hence, the dimensions to build the largest possible athletic complex is 100 by 50 feet
Read more about areas and perimeters at:
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