San Deces – Glass & Mosaics

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
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)
using equation (2)
y=P/2-P/(4+π)-2πP/4(4+π)
2(4+π)P-4P-2πP/4(4+π)
=4P/4(π+4) = P/(π+4)
I have used P for the perimeter. In the last line P=30.
Width of the window = 2(30) / (4+π)= 60 / (4+π) =x
Length of the window = 30/(π+4) = y
Area = xy + π x^2 / 8
Compute the area and multiply by 9.75 to get the total cost.

Note:
d^2A/dx^2 =-1-π/2+π/4 < 0, indicates that the area is maximized.
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