document.write( "Question 1142711: Calculus Optimization\r
\n" ); document.write( "\n" ); document.write( "A closed cylindrical can is to be constructed so that it has a volume of 1 liter (1000 cm3). What are the dimensions that should be used to minimize the amount of material needed to manufacture the can? \n" ); document.write( "

Algebra.Com's Answer #763425 by ikleyn(52776) $\"\"$ $\"About$

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document.write( "As you know, the volume of a cylinder is \r\n" );
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document.write( "    V = , \r\n" );
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document.write( "where pi = 3.14, r is the radius and h is the height.\r\n" );
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document.write( "In your case the volume is fixed:\r\n" );
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document.write( "     = 1000 cubic centimeters.                   (1)\r\n" );
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document.write( "The surface area of a cylinder is \r\n" );
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document.write( "    S =  + ,                               (2)\r\n" );
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document.write( "and they ask you to find minimum of (2) under the restriction (1).\r\n" );
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document.write( "You can rewrite the formula (2) in the form\r\n" );
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document.write( "    S(r) =  + .                         (3)\r\n" );
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document.write( "In formula (3), replace    by  1000, based on (1). You will get\r\n" );
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document.write( "    S(r) =  +  =  + .\r\n" );
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document.write( "The plot below shows the function S(r) =  + , and you can clearly see that it has the minimum.\r\n" );
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document.write( "    \r\n" );
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document.write( "        Plot y =  + \r\n" );
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document.write( "To find the minimum, use Calculus: differentiate the function to get\r\n" );
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document.write( "S'(r) =  +  = \r\n" );
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document.write( "and equate it to zero.\r\n" );
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document.write( "S'(r) = 0   leads you to equation   = ,   which gives \r\n" );
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document.write( "r =  =  = 5.42 cm (approximately).\r\n" );
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document.write( "Answer.  r = 5.42 cm, h =  = 10.84 cm  give the minimum of the surface area.\r\n" );
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