Your answer is "close", so you're doing most of it Right.
You probably made a very simpler error.
A Norman window has the shape of a rectangle surmounted by a semicircle.
If the perimeter of the window is 20ft, find the value of
so that the greatest possible amount of light is admitted.
For my critical number, i get 80/(pi + 4) . It shoud be: .
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The radius of the semicircle is
Hence, the width of the rectangle is
The height of the rectangle is
The perimeter of the semicircle is: .
The perimeter of the rectangle is: .
The total perimeter is 20:
. . .
The area of the semicircle is: .
The area of the rectangle is: .
The total area is: . .
Substitute  into : .
. . And we have: .
Differentiate and equate to zero: .
. . .
Substitute into : .