A window is made up of a circle embedded in a rectangle so that the diameter of the circle is a side of the rectangle. Find the demensions that will admit the most light if the perimeter ofthe window is 44 meters.

Not sure where the calculus comes into play here. The figure will obviously have width equal to twice the height, so that the radius of the semicircle extends all the way to the top of the rectangle.

Any wider, and the semicircle will not fit. Any narrower, and there is unused space above the semicircle.

So, w + w/2 = 22
w = 44/3

The rectangle is 44/3 by 22/3.

Coincidentally, 22/3 = π (almost)