A Norman window consists of a rectangle capped by a semi-circular

A Norman window consists of a rectangle capped by a semi-circular regional. The perimeter of some particular Norman window must be 30 meters. The radius of the semi-circular region is "x" meters. The height of the rectangle is "h" meters. Find the values pf "x" and "h" such that this Norman window has a maximum area. That is, the Norman window will permit the most light to shine through.

3 answers

If the radius is x, then the base of the rectangle is 2x
let the height of the rectangle be h

Perimeter = 30 = (1/2)(2πx) + 2h + 2x
30 = πx + 2h + 2x
h = (30 - πx - 2x)/2

Area = A = 2xh + (1/2)πx^2
= 2x(30 - πx - 2x)/2 + (1/2)πx^2

I will let you finish it ...
simplify a bit, then differentiate,
set the derivative equal to zero and solve for x