Question

A window consisting of a rectangle topped by a semicircle is to have a perimeter 50. Find the radius of the semicircle if the area of the window is to be a maximum

Answer 1

Perimeter equation:
\[2r + 2x + \pi r =50\]
Area:
\[A = 2rx + \frac{\pi r^2}{2}\]
We want to find "r" that maximizes Area
solve perimeter for x
\[x = 25- \frac{\pi+2}{2} r\]
\[A = 50r -2r^2 -\frac{\pi}{2} r^2\]
set derivative equal to 0
\[\frac{dA}{dr} = 50-4r - \pi r = 0\]
\[r = \frac{50}{\pi+4}\]