Question

An architect wants to design a window in the shape of a rectangle capped by a semicircle. If the perimeter of the window is constrained to be 24 feet, what dimensions should the architect choose for the window in order to admit the greatest amount of light?

Answer 1

If the width is x and the rectangle's height is y, then

x+2y + π/2 x = 24
so, y = (24-(1+π/2)x)/2

The area is

a = xy + π/2 (x/2)^2
= x(24-(1+π/2)x)/2 + π/8 x^2
= 12x - (1/2 + π/8)x^2

Now just set da/dx=0 to find where the max occurs. Or, use what you know about the vertex of a parabola.

Answer 2

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