A Norman window has the shape of a rectangle surmounted by a semicircle. (Thus, the diameter of the semicircle is equal to the width of the rectangle.) If the perimeter of the window is 20 ft, find the dimensions of the window so that the greatest possible amount of light is admitted.

Question

My base length is correct: (10/(1+pi/4))
But I need the total height, thanks so much!!

Answer 1

It might be easier if the width of the rectangle is 2x and the height is y, so the perimeter is

2x+2y+πx = 20
y = 10 - (π+2)/2 x

Then the area is

A = π/2 x^2 + 2xy
= π/2 x^2 + 2x(10 - (π+2)/2 x)
= π/2 x^2 + 20x - (π+2)x^2
= -(2 + π/2)x^2 + 20x

The vertex of this parabola is at
x = 20/(4+π)

Note that our answers agree, allowing for the fact that I used 2x for the diameter.

So, y = 10 - (π+2)/2 * 10/(4+π) = 5 + 10/(4+π)