A Norman window has the shape of a rectangle surmounted by a semicircle. The perimeter of the window is 30 ft. The objective is to find the dimensions of the window so that the greatest possible amount of light is admitted. Domain of the model?

See the explanation.

Calling the width of the window ##x## (so the radius of the semicircle is ##x/2##) and the height of the rectangular part of the window ##y##,

We get Area, ##A = pix^2/4+x((30-x-pi/2x)/2)## ##" "## (Rewrite as it pleases you.)

This comes from perimeter ##P = x+2y+pi/2x=30##

Clearly ##0 <= x##, but how do we find the upper bound on ##x##?

The perimeter gives us a linear relationship (with negative slope), so we make ##x## as large as possible by making ##y## as small as possible.

For ##y=0##, we get ##x = 60/(2+pi)##

The domain is ##[0,60/(2+pi)]##