norman window with a perimeter of 50 feet

Question

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 50 feet?

Answer 1

Let x is width and y is height of rectangular part. Then πx/2 + x + 2y = 50; y = 25 - πx/4 - x/2.

Area A = πx²/8 + x(25 - πx/4 - x/2) = 25x - πx²/8 - x²/2; A' = 25 - πx/4 - x = 0; x = 100/(4 + π).ft

Because A'' = - π/4 - 1 < 0 we have maximum

max A = A(100/(4 + π) = 2500/(4 + π) - 10000/(4 +π)²(π/8 + 1/2) = 175.031 ft²