Hi Jason.

A norman window has a semi-circular top to it.

Let r = the radius of the semi-circle.
Let a = the area of the whole window.
Let h = the height of from the sill to the beginning of the semi-circular curve.

So the perimeter is 20 = 2r + 2h + πr.
And the area a = ½ ( πr²) + 2rh.

Solve the first equation for h and substitute this into the second equation - that gets rid of the h and you should get a quadratic in r for the area.

So solve for the maximum value of a, you have the choice of two methods:

1. Calculus: find the derivative with respect to x, set that equal to zero and solve for r. Then plug that value back into the quadratic to find the corresponding value of a.
   
   
2. Parabolic analysis: a quadratic is a parabola, so you can find the maximum by re-arranging the quadratic expression into vertex-graphing form and looking at the vertex of the parabola

Cheers,
Stephen La Rocque