A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 39 ft that will admit the greatest possible amount of light. (Round your answers to two decimal places.)

Answer 1

rectangle height h, width 2r

39 = 2r + 2h + pi r = r(2+pi) +2 h
so
2h = 39 - r(2+pi)

A = area = (1/2) pi r^2+ 2 r h
A = .5 pi r^2 + r[39 -r(2+pi) ]
A = .5 pi r^2 + 39 r - 2r^2 -pi r
A = (.5 pi -2) r^2 +(39-pi)r
dA/dr = 0 for max = 2(.5pi-2) r + (39-pi)
so
r = (39-pi)/(4-pi) etc