Optimizing Window Design: Maximizing Area with Fixed Perimeter

[Noir]

Homework Statement

A window of fixed perimeter is in the shape of a rectangle surmounted by a semi-circle. Prove that its area is greatest when its breadth equals its greatest height.

Homework Equations

SA = lw + (pi*l^2)/4 <--- Thats what I got the surface area to be.
Perimeter = 2w + L(1 + pi / 2)

The Attempt at a Solution

I can solve these problems with numbers, but when it comes to general problems I become unstuck. I tried using the same methord, but it didn't work. Some advice please?

Thanks

 

[lanedance]

you need to maximise the surface area with the perimeter constraint and the easiest way would be to use a lagrange multilpier..

otherwise assume a constant value for the perimeter, say p, solve the perimeter eauqtion for l or w, then substitute back into the SA equation and minimise the function of (now) one variable

 

[Noir]

Cheers, the second methord worked a treat! I'll look into the lagrange stuff, looks interesting :)
Thanks once again.