document.write( "Question 750024: Rectangle QRST between curve $\"y=-x%5E2%2B4\"$ and $\"y=x%5E2-4\"$ . Let P be the point of intersection of the side QT and the x-axis. Let α be the length of the perimeter of this rectangle. We are to find the x-coordinate of the point P where α is maximized and also to find the maximum value of α. P(x,0), where 0\n" ); document.write( "\n" ); document.write( "Therefore, when x=(A), α is maximized and its maximum value is (B)
\n" ); document.write( "solve for A and B
\n" ); document.write( "((this is the picture of the graph http://i44.tinypic.com/30sk9ir.jpg) \n" ); document.write( "

Algebra.Com's Answer #456479 by KMST(5328) $\"\"$ $\"About$

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Both curves and the rectangle QRST are symmetrical with respect to the y-axis.
\n" ); document.write( "TI'll call the coordinates of P (a,0), to distinguish that x-coordinate value $\"a\"$ from the variable $\"x%7D%7D.%0D%0Ahe+x-coordinates+of+points+Q%2C+P+and+T+are+the+same+%7B%7B%7Bx=a\"$ .
\n" ); document.write( "The x-coordinates of points R and S are the same $\"x=-a\"$ .
\n" ); document.write( "The y-coordinate of points Q and R, on curve $\"y=-x%5E2%2B4\"$ is $\"y=-a%5E2%2B4\"$ .
\n" ); document.write( "The y-coordinate of points S and T, on curve $\"y=x%5E2-4\"$ is $\"y=a%5E2-4\"$ .
\n" ); document.write( "The width ST (or QR) of the rectangle is $\"w=2a\"$ .
\n" ); document.write( "The height RS (or QT) of the rectangle is $\"h=-a%5E2%2B4-%28a%5E2-4%29=-a%5E2%2B4-a%5E2%2B4=-2a%5E2%2B8=2%28-a%5E2%2B4%29\"$ .
\n" ); document.write( "The perimeter of the rectangle is
\n" ); document.write( " $\"alpha=2%28w%2Bh%29=2%282a%2B2%28-a%5E2%2B4%29%29=-4a%5E2%2B4a%2B16\"$
\n" ); document.write( " $\"alpha\"$ is a quadratic function in $\"a\"$
\n" ); document.write( "It's maximum is at $\"a=-4%2F%282%28-4%29%29=highlight%281%2F2%29\"$
\n" ); document.write( "because a parabola such as $\"y=ax%5E2%2Bbx%2Bc\"$ has a vertex as $\"x=-b%2F2a\"$
\n" ); document.write( "The equation in vertex form would be
\n" ); document.write( " $\"alpha=-4%28a-1%2F2%29%5E2%2Bhighlight%2817%29\"$
\n" ); document.write( " $\"-4%28a-1%2F2%29%5E2%2B17=-4%28a%5E2-a%2B1%2F4%29%2B17=-4a%5E2%2B4a-1%2B17=-4a%5E2%2B4a%2B16\"$
\n" ); document.write( "So the maximum value of $\"alpha\"$ happens at $\"highlight%28A=1%2F2%29\"$
\n" ); document.write( "and the maximum value of $\"alpha\"$ is $\"highlight%28B=17%29\"$ \n" ); document.write( "

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