A Norman window is a single window that has the shape of a semicircle above a re
ID: 2860365 • Letter: A
Question
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 43 feet. What is the area of the entire window as a function of r and 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 43 feet)? Leave your answer in terms of pi square feet.Explanation / Answer
The window must be a square in order to solve this problem with the given information.
Divide the problem in two parts. Firts you have the square bottom part of the window whose three sides are equal and whose lengths we'll call 'h'. So the perimeter of the square portion is 3h.
The semicircle is the second part. The perimeter of the a circle is 2piR. In this case the diameter of the semicircle equals 'h' so 2piR can be rewriten as 2*pi*h/2 or pi*h. Since this is a semicircle it circumference is one half or pi*h/2.
Area = h^2 + pi * r^2 /2 ----ANSWER a
we have h = 2r
so A(r) = 4r^2 + 1/2 * pi r^2 = r^2( 4+pi/2) -----ANSWER b
The perimeter is
3h + pi*h/2 = 43 feet
4.58*h = 43
h = 43/4.58
= 9.38 ft
The area of the square part is h^2 = 9.38^2 square feet
semicircle pi*(h/2)^2 = 1/2 * pi*(9.38/2)^2
so total area = 9.38^2 +1/2 pi (9.38)^2/4
A = 9.38^2 (1+ pi/8)
A = 87.9844(1+pi/8) -----ANSWER
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