document.write( "Question 1171703: A Norman window consists of rectangle surmounted by a semicircle. If the
\n" ); document.write( "perimeter of a Norman window is 32 ft, what should be the radius of the
\n" ); document.write( "semicircle and the height of the rectangle such that the window will admit most
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Algebra.Com's Answer #796637 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
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document.write( "Let x = the radius of the semicircle, which is half the base of the rectangle.\r\n" );
document.write( "Let h = the height of the rectangle. \r\n" );
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document.write( "The area of the semicircle is half the area of a whole circle.\r\n" );
document.write( "The area of a whole circle circle is\r\n" );
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document.write( "\"Area+=+pi%2Ax%5E2\"\r\n" );
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document.write( "so the area of the semicircle on top is \"expr%281%2F2%29pi%2Ax%5E2\"\r\n" );
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document.write( "The area of the rectangle is \"base%2Aheight=%282x%29%28h%29=2xh\"\r\n" );
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document.write( "So what we want to maximize is the total area y:\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2%2B2xh\"\r\n" );
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document.write( "The perimeter of the entire window is given as 32 feet.\r\n" );
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document.write( "The perimeter (\"circumference\") of the semicircle is half the perimeter (\"circumference\") of a whole circle.\r\n" );
document.write( "The perimeter \"circumference\" of a whole circle is\r\n" );
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document.write( "\"Perimeter+=+%22%27circumference%27%22=+2pi%2Ax\"\r\n" );
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document.write( "so the perimeter (\"circumference\") of the semicircle on top is \"expr%281%2F2%292pi%2Ax\" or \"pi%2Ax\"\r\n" );
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document.write( "The total perimeter is the perimeter of the semicircle plus the left, right,\r\n" );
document.write( "and bottom sides of the rectangle.\r\n" );
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document.write( "\"perimeter+=+pi%2Ax%2Bh%2Bh%2B2x\"\r\n" );
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document.write( "And since the perimeter is 32 feet,\r\n" );
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document.write( "\"perimeter+=+pi%2Ax%2B2h%2B2x=32\"\r\n" );
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document.write( "               \"2h=32-pi%2Ax-2x\"\r\n" );
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document.write( "Substituting for 2h in the equation for the area:\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2%2B2xh\"\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2%2B2h%2Ax\"\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2%2B%2832-pi%2Ax-2x%29%2Ax\"\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2%2B32x-pi%2Ax%5E2-2x%5E2\"\r\n" );
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document.write( "\"y=expr%281%2F2%29pi%2Ax%5E2-pi%2Ax%5E2-2x%5E2%2B32x\"\r\n" );
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document.write( "\"y=%28expr%281%2F2%29pi-pi-2%29x%5E2%2B32x\"\r\n" );
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document.write( "\"y=%28-expr%281%2F2%29pi-2%29x%5E2%2B32x\"\r\n" );
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document.write( "From this point you can get the maximum value by just algebra,\r\n" );
document.write( "using the vertex formula, or by using calculus.  Ikleyn used\r\n" );
document.write( "calculus above so I'll just use algebra.\r\n" );
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document.write( "The x-coordinate of the vertex of a parabola is \"-b%2F%282a%29\", where 'a' is the\r\n" );
document.write( "coefficient of x2, and 'b' is the coefficient of x.\r\n" );
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document.write( "So the x coordinate when the area is a maximum is \r\n" );
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document.write( "\"x=32%2F%28pi%2B4%29\", which is the radius of the semicircle.\r\n" );
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document.write( "Finally we find the height of the rectangle, by substituting \r\n" );
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document.write( "\"32%2F%28pi%2B4%29\" for x in:\r\n" );
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document.write( "\"2h=32-pi%2Ax-2x\"\r\n" );
document.write( "\"2h=32-x%28pi%2B2%29\"\r\n" );
document.write( "\"2h=32-%2832%5E%22%22%2F%28pi%2B4%5E%22%22%29%29%28pi%2B2%29\"\r\n" );
document.write( "Divide every term by 2\r\n" );
document.write( "\"h=16-%2816%5E%22%22%2F%28pi%2B4%5E%22%22%29%29%28pi%2B2%29\"\r\n" );
document.write( "\"h=16-16%28pi%2B2%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=16-%2816pi%2B32%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=16%28pi%2B4%29%2F%28pi%2B4%29-%2816pi%2B32%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=%2816pi%2B64%29%2F%28pi%2B4%29-%2816pi%2B32%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=%28%2816pi%2B64%29-%2816pi%2B32%29%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=%2816pi%2B64-16pi-32%29%2F%28pi%2B4%29\"\r\n" );
document.write( "\"h=32%2F%28pi%2B4%29\"\r\n" );
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document.write( "The radius of the semicircle equals the height. So the rectangle is twice as\r\n" );
document.write( "wide as it is high.  So it looks like this:\r\n" );
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document.write( "Edwin

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