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Question 68456: A Norman window is in the shape of a rectangle surmounted by a semicircle, as shown in the figure. Assume that the perimeter of the window is 32 feet. Express the area of the window as a function of r, the radius of the semicircle. Write your answer using function notation and simplify the function to three terms. Leave pi in its exact form.
I'm having diffficulty setting this problem up could you assist me. Any help is greatly appreciated.
Thanks,
Bridget
Answer by ankor@dixie-net.com(22740)   (Show Source):
You can put this solution on YOUR website! A Norman window is in the shape of a rectangle surmounted by a semicircle, as shown in the figure. Assume that the perimeter of the window is 32 feet. Express the area of the window as a function of r, the radius of the semicircle. Write your answer using function notation and simplify the function to three terms. Leave pi in its exact form.
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We know the width of the window is 2r:
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The area of this shaped window would be:
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Half the area of a circle + area of the rectangle:
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.5(pi*r^2) + (2r * h) (height of the window)
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We have to find the value of h in terms of r, using the given perimeter:
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Circumference is pi*d so half a circumference would be pi*r
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Perimeter equals:
the width + twice the height + half the circumference:
2r + 2h + pi*r = 32
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2h = 32 - 2r - pi*r
2h = 32 - r(2+pi)
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Divide equation by 2:
h = 16 - .5r(2+pi)
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Substitute [16 - .5r(2+pi)] for h in the area equation:
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A = .5(pi*r^2) + (2r * h)
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A = .5(pi*r^2) + 2r[16-.5r(2+pi)]
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A = .5*pi*r^2 + 32r - r^2(2+pi)
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A = .5*pi*r^2 + 32r - 2r^2 - pi*r^2
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A = .5*pi*r^2 - pi*r^2 + 32r - 2r^2; combine like terms
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A = -.5*pi*r^2 - 2r^2 + 32r; should be give the area as a function of r
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Check the perimeter if r = 4, remember 2h = 32 - 2r - pi*r
Find 2h: 2h = 32 - 8 - 12.57 = 11.43
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Perimeter
2r + 2h + pi*r = 32
8 + 11.43 + 12.57 = 32: proves our equation will work for a given r
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