document.write( "Question 399513: why does a square 45x45 have more sqaure footage than a rectangle 30x60. both could be made from the same two 90' pieces \n" ); document.write( "
Algebra.Com's Answer #282909 by Alan3354(69443)\"\" \"About 
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The simplest answer is that 45*45 is more than 30*60.
\n" ); document.write( "For a given perimeter (180 would make a 45 by 45, not 90) the max area of a rectangle is a square.
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\n" ); document.write( "For a perimeter of 180
\n" ); document.write( "P = 2W + 2L = 180
\n" ); document.write( "w + L = 90
\n" ); document.write( "Area = W*L = W*(90 - W)
\n" ); document.write( "To find the max area as a function of W:
\n" ); document.write( "f(W) = w(90 - W) = 90W - W^2
\n" ); document.write( "90W - W^2 - area = 0
\n" ); document.write( "-W^2 + 90W - area = 0
\n" ); document.write( "This is a parabola that is open downward.
\n" ); document.write( "The line of symmetry is W = -b/2a = -90/-2 = 45
\n" ); document.write( "The max is on the line of symmetry:
\n" ); document.write( "f(45) = 90*45 - 45^2 = 4050 - 2025
\n" ); document.write( "Max area = 2025 sq units
\n" ); document.write( "W = 45
\n" ); document.write( "L = 90-45 = 45
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\n" ); document.write( "Any other pair that add to 90 will have a smaller product.
\n" ); document.write( "It will be (45 + x)*(45 - x) = 2025 - x^2
\n" ); document.write( "For any x > 0, it will be less than 2025.
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\n" ); document.write( "Note: 2025 is the max area of a rectangle with a perimeter of 180.
\n" ); document.write( "A circle will have the max area of all figures.
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