What is the area of the largest possible Norman window with a perimeter of 29 feet?

Question

A Norman window has the shape of a semicircle atop a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. What is the area of the largest possible Norman window with a perimeter of 29 feet?

Answer 1

L = 29 = x+2h+πr (πr is half the circumference of a circle)... 2r = x according to the problem so

29 = x +2h +πx/2 , so h = (29 - x - πx/2)/2 and plug that into area

Area = xh + .5*pi*r² = xh + πx²/4... plug in h term and it's now just a function of x, take the derivative and set it equal to zero, get the x value, plug it into Area function (not the derivative of area, but the Area itself)