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Math Help - Finding the height of a minimised cylinder.

  1. #1
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    Finding the height of a minimised cylinder.

    Hi,

    The problem is:
    given a cylinder (I prefer beer can) holding 1000 cubic cm. What should the radius and height be to minimise the surface area.

    I have tried the following
    V=\pi r^2h, V=1000

    therefore,
    h=\frac{1000}{\pi r^2} and S=2\pi r^2+2\pi rh

    subsitituting (h), and simplifying
    S=2 \pi r^2+\frac{2000}{r}

    derive that to get:
    S'=4\pi r-\frac{2000}{r^2}

    Solving for r when S'=0 (critical point), I obtain.
    r=\sqrt[3]{\frac{500}{\pi}}, which according to the text book is correct. Great news. Now I get stuck...

    I cannot get h correct. For some reason h is 2\sqrt[3]{\frac{500}{\pi}}.

    I am so close yet cannot figure out how to get that result.
    Subsituting r into the original equation for h gives:

    \frac{1000}{\pi [\sqrt[3]{\frac{500}{\pi}}]^2}

    This is not correct.

    Thanks
    Craig.
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  2. #2
    Moo
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    Hello,

    You've got the answer


    h=\frac{1000}{\pi \left(\sqrt[3]{\frac{500}{\pi}}\right)^2}

    Now let's do some bits of algebra... remember this property :
    \sqrt[n]{x}=x^{1/n}
    and this : (a^b)^c=a^{bc}


    So h=\frac{2 \cdot 500}{\pi \left(\frac{500}{\pi}\right)^{2/3}}

    h=\frac{2 \cdot 500}{\pi \cdot 500^{2/3} \cdot \pi^{-2/3}}

    h=2 \cdot \frac{500 \cdot 500^{-2/3}}{\pi \cdot \pi^{-2/3}}

    h=2 \cdot \frac{500^{1/3}}{\pi^{1/3}}

    h=2 \sqrt[3]{\frac{500}{\pi}}
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  3. #3
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    Lessons,

    Hi,

    Thanks, these are the school fees. I must learn to persevere.

    Cheers...
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