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. 
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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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