SOLUTION: A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enlosed by the fence?

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Question 335951: A farmer has 1000 feet of fence to enclose a rectangular area. What dimensions for the rectangle result in the maximum area enlosed by the fence?
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
The perimeter of the rectangle is,
P=2L%2B2W=1000
L%2BW=500
The area of a rectangle is,
A=L%2AW
From the perimeter equation,
L=500-W
A=%28500-W%29W=500W-W%5E2
Convert the area function to vertex form (y=a%28x-h%29%5E2%2Bk)to get the maximum value, which occurs at the vertex (h,k).
A%28W%29=500W-W%5E2=-%28W%5E2-500W%29
A%28W%29=-%28W%5E2-500W%2B62500%29%2B62500
A%28W%29=-%28W-250%29%5E2%2B62500
The maximum area occurs when W=250ft and is equal to A=62500.
L=500-250
L=250ft
The maximum area for a given perimeter is a square.