A Norman Window with Maximum Light?

i = incident light per unit area of the window

As = Area of semicircle = (1/2)pi*r^2

Ls = light admitted by semicircle = As*i/2 = (1/4)pi*r^2*i

Ar = area of rectangle = 2*r*h

Lr = light admitted by rectangle = Ar*i = 2*r*h*i

L = Lr + Ls = 2*r*h*i + (1/4)pi*r^2*i

P = pi*r + 2*(r + h)

h = (P - pi*r - 2*r)/2 = [P - r(pi + 2)]/2

Since "i" is just some constant I can set it equal to 1 (it comes out anyway when you take the derivative and set it equal to 0 for the maximum L).

L = r*(P - pi*r - 2*r)+ (1/4)pi*r^2

L = P*r - (3/4)pi*r^2 - 2*r^2

dL/dr = 0 = P - (3/2)pi*r - 4*r

r = P/[4 + (3/2)pi] = 2*P/[8 + 3*pi]

Width = 2*r = 4*P/[8 + 3*pi] = 0.23*P

Height = [P - r(pi + 2)]/2 = [P - 2*P(pi + 2)/(8 + 3*pi)]/2

Height = (P/2)[1 - 2*(pi + 2)/(8 + 3*pi)] = 0.205*P

My width agrees with that other answer.

there's a much less complicated way, whether it seems that that some "wager & verify" may be the thank you to bypass (except you be responsive to 3 calculus). i could %. some lengths and widths of the window to confirm what is going to maximize the component to the rectangle, then compute the component to the semi-circle in keeping with my effects. the component to the circle is a function, the bigger it is the greater mild will are available in.

What I get is 2r = 4P / (3π + 8)