document.write( "Question 316180: A volume is defined by the function V(h)=h(h-6)(h-12). What is the maximum volume for the domain 0\n" ); document.write( "

Algebra.Com's Answer #226143 by Fombitz(32388) $\"\"$ $\"About$

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$\"V=h%28h-6%29%28h-12%29\"$
\n" ); document.write( " $\"V=h%28h%5E2-18h%2B72%29\"$
\n" ); document.write( " $\"V=h%5E3-18h%5E2%2B72h\"$
\n" ); document.write( "

\n" ); document.write( "The maximum occurs between $\"x=2\"$ and $\"x=3\"$ .
\n" ); document.write( "You can find the maximum by taking the derivative of the function and finding when it equals zero. The value of the derivative is also the slope of the tangent line to the function.
\n" ); document.write( " $\"dV%2Fdh=3h%5E2-18h%2B72\"$
\n" ); document.write( " $\"dV%2Fdh=3%28h%5E2-12h%2B24%29=0\"$
\n" ); document.write( " $\"h%5E2-12h%2B24=0\"$
\n" ); document.write( "Use the quadratic formula,
\n" ); document.write( " $\"h=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"$
\n" ); document.write( " $\"h=+%28-%28-12%29+%2B-+sqrt%28+144-4%2A1%2A24+%29%29%2F%282%2A1%29+\"$
\n" ); document.write( " $\"h=+%2812+%2B-+sqrt%28+144-96%29%29%2F%282%2A1%29+\"$
\n" ); document.write( " $\"h=+%2812+%2B-+sqrt%28+48%29%29%2F2+\"$
\n" ); document.write( " $\"h=+%2812+%2B-+4sqrt%28+3%29%29%2F2+\"$
\n" ); document.write( " $\"h=+6+%2B-+2sqrt%28+3%29+\"$
\n" ); document.write( "The maximum occurs at $\"h=6+-2sqrt%28+3%29+\"$ .
\n" ); document.write( "The minimum occurs at $\"h=6+%2B2sqrt%28+3%29+\"$ .
\n" ); document.write( " $\"h=6+-2sqrt%28+3%29=2.536+\"$
\n" ); document.write( " $\"Vmax=2.536%282.536-6%29%282.536-12%29\"$
\n" ); document.write( " $\"Vmax=83.1348\"$
\n" ); document.write( "The maximum volume is 83 cubic feet. \n" ); document.write( "

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