Imagine a fenced-in area composed of a rectangle with one side being a semicircle.
The perimeter of this fence is 498 feet.
Maximize the area of this pen.
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The radius of the semicircle is
Hence, the length of the rectangle is
Let the width of the rectangle be
The semicircle has perimeter
The rectangular portion of the pen has perimeter
We have: .
The area of the semicircle is: .
The area of the rectangle is: .
The area of the pen is: .
Substitute  into : .
. . which simplifies to: .
And that is the function we must maximize . . .