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?

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See the interpretation.

Calling the width of the window ##x## (so the radius of the semicircle is ##x/2##) and the culmination of the athwart bisect 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 confront the upper jump on ##x##?

The perimeter gives us a straight homogeneity (delay privative climb), so we execute ##x## as capacious as likely by making ##y## as trivial as likely.

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

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

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