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Math Help - maximize volume, minimize surface area

  1. #1
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    maximize volume, minimize surface area

    Given a volume of 1700 cm^3, minimize the surface area of a cylinder.

    S= 2 \pi rh+2 \pi r^2

    V= \pi r^2h=1700

    so, for substitutions:
    h=1700/ \pi r^2, which I can put into the S equation to get:
    S=2 \pi r (1700/r^2) +2 \pi r^2

    And I think I can simplify the above equation to:
    3400 \pi /r +2 \pi r^2

    and so is the derivative of this equal to:
    -3400 \pi /r^2 +4 \pi r

    I wasn't completely sure if the above equation was right. But continuing on, I would set the derivative equal to 0 and solve for r.

    Any help would be greatly appreciated!-I know this is a long post, sorry.
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  2. #2
    MHF Contributor Unknown008's Avatar
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    You missed the pi there:

        S=2 \pi r (1700/r^2) +2 \pi r^2

    It should be:

        S=2 \pi r \left(\dfrac{1700}{\pi r^2}\right) +2 \pi r^2
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  3. #3
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    I can't believe I missed that! Thank you. So to simplify that equation:

    3400/r + 2 \pi r^2

    So the new derivative is:
    -3400/r^2 +4 \pi r

    and using my graphing calc, there is a zero at 6.4677966...I guess thats right
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  4. #4
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    so I assume the above value I got for r is the minimum? then I just substitute it into the other equation and solve for h...right?
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  5. #5
    MHF Contributor Unknown008's Avatar
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    Yes, but why 'assume' while you can be sure?

    Remember that if \dfrac{d^2S}{dr^2} > 0, the stationary point is a minimum.
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