Please answer the following A Norman window is a single window that has the shap

Question

Please answer the following

A Norman window is a single window that has the shape of a semicircle above a rectangle so that the diameter of the semicircle is equal to the width of the rectangle. Suppose that a Norman window has an outside perimeter of 27 feet. what is the area of the entire window as a function of r and h, where r is the radius of the semicircular part and h is the height of the rectangular part? Area square feet. What is the area of the entire window as a function of r only? A(r) square feet. what is the largest possible area of such a Norman window (one with outside perimeter 27 feet? Leave your answer in terms of TT ANSWER square feet.

Explanation / Answer

Let the height of the rectangle be h and the radius of the semicircle be r.

Area of rectangle = (2r)*h = 2rh
Area of semicircle = 1/2 pi r^2
So we want to maximize A = 2rh + 1/2 pi r^2
but there are two variables, so we'll first try to eliminate one of them.

Note that perimeter = 2h + 2r + pi r = 27
so h = 27 - r - pi/2 r = 27- (1 + pi/2)r

Substitute that into our equation for area to get:
A = 2r(27-(1+pi/2)r) + 1/2 pi r^2
= 54r - 2r^2 - 1/2 pi r^2

Take the derivative:
dA/dr = 54 - 4r - pi r
and set it equal to zero:
0 = 54 - 4r - pi r
(4 + pi) r = 54
r = 54 / (4 + pi)

Now that you have the radius, you can find the area A by using the equation
A = 54r - 2r^2 - 1/2 pi r^2

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