document.write( "Question 1155489: A rectangular cardboard poster is to have a 96 - square - inch rectangular section of printed material, a 2 - inch border top and bottom, and a 3 - inch border on each side. Find the dimensions and area of the smallest poster that meets these specifications. (Note; Let x and y be the dimensions of the 96 - square - inch area.) \n" ); document.write( "

Algebra.Com's Answer #778070 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "They want you to minimize\r\n" );
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document.write( "    f(x,y) = (x-6)*(y-4)  under the condition  x*y = 96.  (1)\r\n" );
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document.write( "where x and y are the dimensions of the poster.\r\n" );
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document.write( "In other words, from (1), you need to minimize\r\n" );
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document.write( "    f(x,y) = xy - 4x - 6y + 24 = 96 - 4x - 6y + 24 = -4x - 6y + 120.\r\n" );
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document.write( "From (1), you have  y = , so the function f(x,y) takes the form\r\n" );
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document.write( "    g(x) = -4x -  + 120 = -4x -  + 120.    (2)\r\n" );
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document.write( "So, you differentiate (2), and you get\r\n" );
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document.write( "    g'(x) = -4 + .\r\n" );
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document.write( "Equate it to zero\r\n" );
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document.write( "    g'(x) = 0 = -4 + .\r\n" );
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document.write( "So, to find x, you have this equation\r\n" );
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document.write( "     = 576,\r\n" );
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document.write( "     =  = 144\r\n" );
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document.write( "     x =  = 12 centimeters.\r\n" );
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document.write( "Then y =  = 8 cm.\r\n" );
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document.write( "Thus the dimension of the poster are  12 inches width and 8 inches height.\r\n" );
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