A Norman window has the shape of a semicircle atop a rectangle
so that the diameter of the semicircle is equal to the width of the rectangle.
What is the area of the largest possible Norman window with a perimeter of 45 feet?
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The radius of the semicircle is
Then the width of the rectangle is
Let = height of the rectangle.
The perimeter is: . . 
The area of the window is: . . 
Substitute  into : .
. . which simplifies to: . . 
Differentiate and equate to zero: .
. . Hence: .
Substitute into : .
. . and we get: .