Question

a norman window is constructed by adjoining a semicircle to the top of an ordinary rectangular window. find the dimensions of a norman window of maximum area if the total perimeter is 72 units. Base value is 21.

Answer 1

if the rectangle is x wide by y high, then the diameter of the window is x.

So
x + 2y + π/2 x = 72
thus, y = [72-(1 + π/2)x]/2

The area is
a = xy + π/8 x^2
= x[72-(1 + π/2)x]/2 + π/8 x^2
= 36x - (1+π/4)x^2 + π/8 x^2
= 36x - (1 + 3π/8)x^2

da/dx = 36 - (2 + 3π/4)x
da/dx=0 when
x = 36/(2+ 3π/4)) = 144/(8+3π)
now you can easily find y.
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right?