Question

A window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the
window is 30ft, express the area of the window as a function of the width of the window.

Answer 1

let the width of the window be x
Let the length of the window be y, which makes
the radius of the semicircle y/2

perimeter = 2x + y + πy = 30
y(1 + π) = 30-2x
y = (30 - 2x)/(1 + π)

area = xy + (1/2)π r^2
= x(30-2x)(1+π) + (1/2)π([(30-2x)/(1+π)]^2 / 4

Suppose we define the width as x and the length as 2y, which makes
the radius of the half-circle as y
then we get
2x + 2y + 2πy = 30
x + y + πy = 15
y(1+π) = 15-x
y = (15-x)/(1+π)

area = 2xy + (1/2)π y^2
= 2x(15-x)/(1+π) + (1/2)π [(15-x)/(1+π)]^2

take your pick, but check my algebra for each