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The Hammer Window

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What is the Question? A Hammer Window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 feet, find the dimensions of the window so that the greatest possible amount of light is admitted.

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We have a window that looks something like this: We must maximize the area of the window, while staying within the perimeter requirement. What Does That Mean? ll w r

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How Do We Do That? We took the perimeter formula for both the rectangle and the circle to come up with this unique Hammer Window Perimeter Formula: P = w + 2 l + p r w = width of Hammer windowdomain of w : (0,30) l = length of Hammer windowdomain of l : (0,30) r = radius of semicircle on top w = 2 r or r = ½ w p = 3.142P = 30

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How Do We Do That? We solved for l and substituted it into our special Hammer Window Area formula: A = lw + ½ p r 2 So now, we solve it!

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Solving These Suckers! P = w + 2 l + p r w = 2 r 30 = 2 r + 2 l + 3.142 r 30 - 5.142 r = 2 l (30 - 5.142 r ) ½ = l 15 - 2.571 r = l A = lw + ½ p r 2 A( r ) = (15-2.571 r )2 r + ½(3.142) r 2 A( r ) = 30 r - 5.142 r 2 + 1.571 r 2 A( r ) = 30 r - 3.571 r 2 A( r ) = 30 - 7.142 r

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Maximize Area A( r ) = 30 - 7.142 r 0 = 30 - 7.142 r 7.142 r = 30 r = 4.2 w = 2 r w = 8.4 Classify r = 4.2 as a maximum: Look at the graph of A( r ) = 30 r - 3.571 r 2 and see that it achieves its maximum value at 4.2

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The Graph (4.2, 63) A( r ) = 30 r - 3.571 r 2

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So What is the Length? Substitute r into Hammer Window Perimeter formula: P = w + 2 l + p r P = 2 r + 2 l + p r 30 = 2(4.2) + 2 l + 3.142(4.2) 30 = 8.4 + 2 l + 13.19 30 = 2 l + 21.59 8.41 = 2 l 4.2 = l And the pieces come together...

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In English... What the heck does all that junk mean??? Wellllll…….. It means that the Hammer Window is 8.4 wide. The length of the rectangle section is 4.2, and the length of the semicircle on top is equal to the radius, which is 4.2. Overall, the length is 8.4.

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The Hammer Window r = 4.2 w = 8.4 Overall l = 8.4 l = 4.2

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Dr. Hammer is the coolest teacher in the WORLD!

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MERRY CHRISTMAS!!! and a happy Halloween!and a happy Halloween! the end

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