document.write( "Question 1150956: A rectangular area of 1,050 square feet is to be enclosed by a fence, then divided down the middle by another piece of fence. The fence down the middle costs $0.50 per running foot, and the other fence costs $1.50 per running foot. Find the minimum cost for the required fence. \n" ); document.write( "

Algebra.Com's Answer #772584 by ikleyn(52793) $\"\"$ $\"About$

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document.write( "From the context, the dimensions of the rectangle are not given for advance - they are unknowns and they should be found\r\n" );
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document.write( "from the minimum cost condition.\r\n" );
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document.write( "Let x be one dimension and y be the other dimension of the rectangle.\r\n" );
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document.write( "Then the cost of the outside perimeter fence is 1.50*(2x+2y) dollars = 3*(x+y) dollars,\r\n" );
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document.write( "while the cost of the fence down middle is 0.50*x dollars.\r\n" );
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document.write( "    Note, that I don't know now, which dimension will be the length and which be the width.\r\n" );
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document.write( "    When the problem will be solved, the solution will tell me it . . . \r\n" );
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document.write( "So, I need minimize the function  \r\n" );
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document.write( "        f(x,y) = 3*(x+y) + 0.5x     (1)\r\n" );
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document.write( "under the condition\r\n" );
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document.write( "        x*y = 1050.                 (2)\r\n" );
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document.write( "From (2), express  y =   and substitute it into (1). You will get\r\n" );
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document.write( "        g(x) =  =  +  + .\r\n" );
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document.write( "Differentiate it over x\r\n" );
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document.write( "        g'(x) =  -  + \r\n" );
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document.write( "and equate the derivative to zero.  You will get\r\n" );
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document.write( "        3.5 = ,    or\r\n" );
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document.write( "        3.5x^2 = 3150\r\n" );
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document.write( "           x^2 =  = 900,\r\n" );
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document.write( "           x   =  = 30.\r\n" );
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document.write( "Thus the dimensions of the rectangle are 30 ft  and   = 35 ft.\r\n" );
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document.write( "The fence down middle has the length of 30 ft;  hence, it is parallel to the shorter side of the rectangle.\r\n" );
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\n" ); document.write( "\n" ); document.write( "If you want to see many other similar solved problems, look into the lesson \r
\n" ); document.write( "\n" ); document.write( " - Calculus optimization problems\r
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