Answer to Question #134480 in Calculus for monibangs

Question

A Normal window with a perimeter of 20 units has the shape of a rectangle surmounted by a
semi-circle. If its base is x units long and the height of the rectangle is y units long, write a mathematical model for its area in terms of x.

Answer 1

Radius of semicircle$=\frac{x}{2}$

Perimeter of window$=x+2y+\frac{πx}{2}=20$

$y=10-\frac{x}{4}(2+π)$

Area of window$=xy+\frac{π}{2}\times\frac{x^2}{4}$

Substituting $y$ in area, we get

Area of window$=10x-\frac{x^2}{4}(2+\frac{π}{2})$