Designing a norman window

Question

A Norman window has the shape of a rectangle surmounted by a semicircle. Find the dimensions of a Norman window of perimeter 29 ft that will admit the greatest possible amount of light. (Round your answers to two decimal places.)

A rectangle surmounted by a semicircle is shown. The flat side of the semicircle is shared with the top side of the rectangle, half of this side is labeled x. The height of the rectangle is labeled y.

X= ? ft

Y= ? ft

Answer 1

Perimeter = 2x + 2y + πx = 29

Area = 2xy + 0.5πx²

From perimeter y = (29-2x-πx) / 2

Substituting

A = area = 0.5πx² + 2x (29-2x-πx)/2

A = x² (0.5π -2-π) + x (29)

dA/dx = 2x (0.5π-2-π) + 29

= 2x (0.5π -2) + 29

We want dA/dx = 0, so you can just solve that