document.write( "Question 1366: The owner of a cannery wants to design a can with a volume of 300 cubic centimeters. He wants the can to be made of the least amount of material in an effort to minimize the cost of manufacturing. Find the radius and height of the can that would satisfy his requirements. \n" ); document.write( "

Algebra.Com's Answer #411 by AnlytcPhil(1806) $\"\"$ $\"About$

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The owner of a cannery wants to design a can with a volume of 300 cubic
\n" ); document.write( "centimeters. He wants the can to be made of the least amount of material in an
\n" ); document.write( "effort to minimize the cost of manufacturing. Find the radius and height of the
\n" ); document.write( "can that would satisfy his requirements.
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We are asked to minimize the surface area of a closed right circular cylinder,\r\n" );
document.write( "We need two formulas, one for the volume and one for the surface area of a\r\n" );
document.write( "closed cylinder. These are  \r\n" );
document.write( "V = pr²h and A = 2prh+2pr²\r\n" );
document.write( "Since the volume is fixed at 300 cm³, we have\r\n" );
document.write( "300 = pr²h\r\n" );
document.write( "Solve this for the height h\r\n" );
document.write( "h = 300/(pr²)\r\n" );
document.write( "Now substitute 300/(pr²) for h in the surface area formula\r\n" );
document.write( "A = 2prh+2pr² \r\n" );
document.write( "A = 2pr[300/(pr²)] + 2pr²\r\n" );
document.write( "A = 600/r + 2pr²\r\n" );
document.write( "Now we take the derivative dA/dr\r\n" );
document.write( "dA/dr = -600/r² + 4pr\r\n" );
document.write( "set this = 0\r\n" );
document.write( "-600/r² + 4pr = 0\r\n" );
document.write( "Multiply through by LCD = r²\r\n" );
document.write( "-600 + 4pr³ = 0\r\n" );
document.write( "4pr³ = 600\r\n" );
document.write( "  r³ = 600/(4p)\r\n" );
document.write( "  r³ = 150/p

\n" ); document.write( "` ` ` ``_____
\n" ); document.write( "` r = ³Ö150/p = 3.627831679 cm, approximately.


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document.write( "  h = 300/(pr²) = 7.255663357 cm, approximately
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document.write( "So the height is twice the radius, making the diameter equal to the height.
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document.write( "If the can were sliced in half vertically the cross section would be a square.
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document.write( "Edwin\r
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