Question

A Norman window with outer perimeter 20 ft is to be constructed.

h = length of the rectangle
r is radius of the semi circle


Express h in terms of r.

Express the area of the window in terms of the radius, r.

Answer 1

so.. if you check the picture below, the "width" of the rectangle, is really just 2r

thus
`\bf \textit{perimeter of the semi-circle}\\\\ P_s=\cfrac{2\pi r}{2}\to \pi r \\\\\\ \textit{perimeter of the rectangle}\\\\ P_r=h+h+2r\to 2(h+r) \\\\\\-----------------------------\\\\ \textit{perimeter of the window}\\\\ P_w=P_s+P_t\implies 20=\pi r+2(h+r)\implies 10-\cfrac{\pi r}{2}=h+r \\\\\\ \boxed{10-\cfrac{\pi r}{2}-r=h}`

now.. as for the Area of it


`\bf \textit{area of the semi-circle}\\\\ A_s=\cfrac{\pi r^2}{2} \\\\\\ \textit{area of the rectangle}\\\\ A_r=2hr\implies 2\left( 10-\cfrac{\pi r}{2}-r \right)r\implies 20r-\pi r^2-2r^2\\\\ -----------------------------\\\\ \textit{area of the window}\\\\ A_w=A_s+A_r\implies \boxed{A_w=\left( \cfrac{\pi r^2}{2} \right)+\left( 20r-\pi r^2-2r^2 \right)}`