document.write( "Question 1013607: Paul wants to enclose part of his yard to make a rectangle dog run.
\n" ); document.write( "He will put up a fence on the three sides of the rectangle and use
\n" ); document.write( "the wall of the house as the fourth side of the rectangle. If he has
\n" ); document.write( "36 yd of fencing material, and wants the dog run to be the maximum
\n" ); document.write( "are possible, what length should make the dog run? Assume that the
\n" ); document.write( "length is longer than the width. \n" ); document.write( "

Algebra.Com's Answer #630462 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "Paul wants to enclose part of his yard to make a rectangle dog run. \r\n" );
document.write( "He will put up a fence on the three sides of the rectangle and use \r\n" );
document.write( "the wall of the house as the fourth side of the rectangle. If he has\r\n" );
document.write( "36 yd of fencing material, and wants the dog run to be the maximum \r\n" );
document.write( "are possible, what length should make the dog run? Assume that the \r\n" );
document.write( "length is longer than the width.\r\n" );
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document.write( "We assume there is no fencing required along the red line,\r\n" );
document.write( "the side of the house.  Let x be the width of the rectangle and L be \r\n" );
document.write( "the length of the rectangle.  Then the Area, y, is given by\r\n" );
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document.write( "(1)     y = Lx\r\n" );
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document.write( "Since we are told that Paul has 36 yd. of fencing, then\r\n" );
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document.write( "x+L+x = 36 or 2x+L = 36\r\n" );
document.write( "                  L = 36 - 2x\r\n" );
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document.write( "We substitute that in equation (1):\r\n" );
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document.write( "        y = Lx\r\n" );
document.write( "        y = (36 - 2x)x\r\n" );
document.write( "        y = x(36 - 2x)\r\n" );
document.write( "        y = 36x - 2x²\r\n" );
document.write( "        y = -2x² + 36x\r\n" );
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document.write( "This is the equation of a parabola that opens downward:\r\n" );
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document.write( "Since the area is y, we need to find what point has the highest\r\n" );
document.write( "possible value for y.  That will be at the highest point on the\r\n" );
document.write( "parabola. That highest point is the vertex.  So we use the \r\n" );
document.write( "vertex formula:\r\n" );
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document.write( "x-coordinate of vertex of parabola y = ax²+bx+c is -b/(2a)\r\n" );
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document.write( "We compare y = -2x² + 36x to\r\n" );
document.write( "           y = ax² + bx + c and find that a=-2, b=120 and c=0\r\n" );
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document.write( "so the x-coordinate of the vertex is -36/[2∙(-2)] = -36/(-4) = 9\r\n" );
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document.write( "So the maximum area will occur when x = 9, which will be the width\r\n" );
document.write( "of the rectangle, and since the length is given by  L = 36 - 2x\r\n" );
document.write( "the length will be  L = 36 - 2(9) = 36 - 18 = 18.\r\n" );
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document.write( "Therefore the solution is to make the width be 9 yd. and the length \r\n" );
document.write( "be 18 yd.\r\n" );
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document.write( "Edwin

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