Optimization Help and explanation

Question

A Norman window is shaped like a rectangle with a semicircle on top of it. A carpenter has 10m of wood trim and wishes to cover the sides of the rectangle and the upper part of the semicircle. Which dimensions of the rectangle will maximize the area of the window? What is that area?

Answer 1

The total area of the figure (let w be the width of the rectangle and diameter of the semicircle and h be the height of the rectangle)

A = (1/2)(π(w/2)²) + hw = (1/8)πw² + hw

The perimeter of the body is constrained to 10 meters:

P = π(w/2) + w + 2h = 10

Use the constraint to put h in terms of w: h = 1/2 (10-w-πw/2) = 5-w/2-πw/4

Substitute into A expression to get A(w)

A = (1/8)πw² + w(5 - w/2 - πw/4)

Set dA/dw = 0

πw/4 + 5 - w - πw/2 = 0

Solve for w, h, A.