Question

A unique window has the shape of a rectangle with a quarter circle on top. What height, width, and radius
will maximize the enclosed area if the outside perimeter is 4 m?

Answer 1

Assuming the radius of the 1/4 circle is the width of the rectangle, then if the rectangle has height h, the perimeter is
p = π/2 r + r + h + r + h = 2h + (2 + π/2)r = 4
so h = 4 - (2 + π/2)r
The area is
a = hr + 1/4 πr^2
= (4 - (2 + π/2)r)*r + 1/4 πr^2
= 4r - (2 + π/4) r^2
da/dr = 4 - (4 + π/2)r
da/dr=0 when r = 8/(π+8)
now just find h. The width is the same as the radius r.