document.write( "Question 1137049: An open box with a square base is to have a volume of 108 cubic inches. Find the dimensions of the box that will have minimum surface area. \r
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Algebra.Com's Answer #754881 by ikleyn(52777) $\"\"$ $\"About$

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document.write( "The volume equation is  \r\n" );
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document.write( "    x^2*h = 108.     (1)\r\n" );
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document.write( "The surface area expression for the open box is  \r\n" );
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document.write( "    A(x,h) = x^2 + 4xh.     (2)\r\n" );
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document.write( "So, you need to find dimensions which minimize the function A(x,h) (2)  under the condition (1).\r\n" );
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document.write( "To solve the problem, express h =  from (1) and substitute it into (2), making A function of only one variable x:\r\n" );
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document.write( "    A(x) = x^2 + 4x* = x^2 + .    (3)\r\n" );
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document.write( "Now you have this function A(x) of one variable x, and you should find its minimum.\r\n" );
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document.write( "Differentiate; equate the derivative to zero\r\n" );
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document.write( "    A'(x) = 2x -  = 0\r\n" );
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document.write( "and get\r\n" );
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document.write( "    2x^3 - 432 = 0  ====>  x^3 = 432/2 = 216  ====>  x =  = 6.\r\n" );
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document.write( "Answer.  x= 6;  h =  =  =  = 3.\r\n" );
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