The area of the pen is the products of its dimensions
Let the dimension of the fence be x by y.
So, we have:
[tex]\mathbf{2x + 5y = 1200}[/tex] --- perimeter
[tex]\mathbf{Area =xy}[/tex] -- area
Subtract 5y from both sides of [tex]\mathbf{2x + 5y = 1200}[/tex]
[tex]\mathbf{2x = 1200 - 5y}[/tex]
Divide both sides by 2
[tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex]
Substitute [tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex] in [tex]\mathbf{Area =xy}[/tex]
[tex]\mathbf{Area = \frac{1200 - 5y}{2} \times y}[/tex]
[tex]\mathbf{Area = \frac{1200y - 5y^2}{2}}[/tex]
Split
[tex]\mathbf{Area = 600y - \frac{5}{2}y^2}[/tex]
Differentiate
[tex]\mathbf{A' = 600 -5y}[/tex]
Set to 0
[tex]\mathbf{600 -5y = 0}[/tex]
Add 5y to both sides
[tex]\mathbf{5y = 600}[/tex]
Divide both sides by 5
[tex]\mathbf{y = 120}[/tex]
Substitute [tex]\mathbf{y = 120}[/tex] in [tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex]
[tex]\mathbf{x = \frac{1200 - 5 \times 120}{2}}[/tex]
[tex]\mathbf{x = \frac{1200 - 600}{2}}[/tex]
[tex]\mathbf{x = \frac{600}{2}}[/tex]
[tex]\mathbf{x = 300}[/tex]
Recall that:
[tex]\mathbf{Area =xy}[/tex]
So, we have:
[tex]\mathbf{Area = 300 \times 120}[/tex]
[tex]\mathbf{Area = 36000}[/tex]
Hence, the maximum area of the pen is 36000 square feet.
Read more about maximum areas at:
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