document.write( "Question 1148825: A closed rectangular container with a square base is to have a volume of 18000 inches^3 . The material of the top and bottom of the container will cost $2 per inch^2 , the material for the sides will cost $3 per inch^2 . Find the dimensions of the container of least cost. \n" ); document.write( "

Algebra.Com's Answer #770163 by ikleyn(52787) $\"\"$ $\"About$

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document.write( "Let x be the length of the side of the base, and\r\n" );
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document.write( "let y be the height of the container.\r\n" );
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document.write( "Then the volume is   = 18000 cubic inches.\r\n" );
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document.write( "The cost of the material for two bases is   dollars;\r\n" );
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document.write( "the cost of the material for four lateral sides is  4*3*x*y = 12xy dollars.\r\n" );
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document.write( "The problem requires us to minimize   = 18000.\r\n" );
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document.write( "Using the restriction formula, express  y =   and substitute it  into the total cost expression.\r\n" );
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document.write( "Then the cost takes the form  C(x) =  = .\r\n" );
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document.write( "Thus the problem is just reduced to finding minimum of the function\r\n" );
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document.write( "    C(x) = .\r\n" );
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document.write( "To find it minimum, take a derivative and equate it to zero.  It gives you the equation\r\n" );
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document.write( "    C'(x) =  = 0,\r\n" );
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document.write( "which implies\r\n" );
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document.write( "    4x^3 = 216000,\r\n" );
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document.write( "     x^3 = 54000\r\n" );
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document.write( "     x =  = .\r\n" );
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document.write( "Then  y =  =  = .\r\n" );
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document.write( "ANSWER.  x=  = 37.798 inches (approximately)  and  y=  = 12.6 inches (approximately).\r\n" );
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document.write( "PARTIAL CHECK.   =  = 18001 in^3.  The miserable difference is due to rounding.\r\n" );
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