A window in the shape of a rectangle capped by a semicircle is to be surrounded by 150 inches of molding. Find the radius of the semicircle part if the total area of the window is to be a maximum.

Question

Optimization problem

Answer 1

If the radius of the semicircle is R, then the side of the rectangle is 2R by a, where a is the other side of the rectangle. The perimeter of the window is (pi)R+2R+2a=150, because the moulding is fixed around the perimeter. The rectangular part of the window has the most effective area when the rectangle is a square, so a=2R.

We can write (pi)R+6R=150, so R((pi)+6)=150, so R=150/((pi)+6)=150/9.142=16.41 inches.