Optimization & Lagrange Multipliers · Hard problem

A stained-glass window is shaped like a rectangle of width $2r$ and height $h$, topped by a semicircle of radius $r$ (so the semicircle's diameter is the rectangle's top side). The window must enclose exactly $50$ square feet of glass, and only the two vertical sides, the bottom, and the curved top edge need a metal frame (there is no frame along the straight line where the rectangle meets the semicircle). Find the minimum possible length of framing, in feet. Give an exact answer.