how do this problem

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 20 feet? 

Answer 1

You are maximizing area so write an equation for the area and then take the derivative. Then set the derivative equal to zero to solve for the local maximum.

Area_{circular portion} = (1/2)πr²

Area_{rectangular portion} = L•W

So A = (1/2)πr² + L•W

But, because "the diameter of the semicircle is equal to the width of the rectangle" then 2r = W

And because "the perimeter" is 20, then πr + 2L + W = 20

You can now substitute into the area equation to get the area as a function of a single variable.

Take the derivative, set it equal to zero, and solve.