### A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288in. find the radius of the semicircle that will maximize the area of the window?

Let x be the width and y be the length of the rectangle. x/2 is
the radius of the semicircle Perimeter of the Norman window is
x+2y+(π x)/2 Let P be the perimeter --- 288 in this problem. P =
x+2y+(π x)/2--------(1) Solving for y from equation (1) 2y =
P-x-πx/2 y = P/2-x/2-πx/4--------(2) Area = xy + π x^2 / 8 A =
x(P/2-x/2-π x/4) + π x^2/8 A= Px/2-x^2 /2 -πx^2/4 +πx^2/8 dA/dx =
P/2 -2x/2-2πx /4 +2πx / 8 =0 (4p-8x-2πx)/8=0 4p-2x(π+4)=0
4p=2x(π+4) x= 2P / (4+π) The radius is x/2 = P/(4+PI) Substitute P
with 288 radius = 288 / (4+PI) will maximize the area of the
window.
d^2A/dx^2 =-1-π/2+π/4 < 0, indicates that the area is
maximized. You'll have to simplify x and y if you want them in
numeric format.