Precal: : Maximum

Question

A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular Norman window is to be 24 feet. What should its dimensions be in order to maximize the area of the window and, therefore, allow in as much light as possible?

Answer 1

Let 2y is diameter of semicircle and x is other side of rectangular part. Then area of window S = 2xy + πy²/2

and perimeter 2y + 2x + πy = 24; 2x = 24 - 2y - πy; S = 24y - 2y² - πy² + πy²/2 = 24y - (2 + π/2)y² =

-(2 + π/2)[y - 12/(2 + π/2)]² + 144/(2 + π/2);

If y = 12/(2 + π/2) = 3.36 ft and x = 12 - y(1 + π/2) = 12 - 3.36(1 + π/2) 3.36 ft