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Math Help - Optimization Cylinder

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    Optimization Cylinder

    An open-topped cylindrical glass jar is to have a given capacity. Find the ratio of height to diameter if the area of glass is a minimum.
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  2. #2
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    Quote Originally Posted by robertosavin View Post
    An open-topped cylindrical glass jar is to have a given capacity. Find the ratio of height to diameter if the area of glass is a minimum.
    You need surface area in terms of a single variable, in order to easily differentiate,
    so we can write h "in terms of" r.

    The jar volume is V={\pi}r^2h\ \Rightarrow\ h=\frac{V}{{\pi}r^2}

    It's surface area is {\pi}r^2+2{\pi}rh={\pi}r^2+\frac{2{\pi}rV}{{\pi}r^  2}={\pi}r^2+\frac{2V}{r}

    To find the minimum surface area, differentiate surface area with respect to r
    and set it to zero.

    \frac{d}{dr}\left({\pi}r^2+2Vr^{-1}\right)=2{\pi}r-\frac{2V}{r^2}

    If this is zero then

    2{\pi}r=\frac{2V}{r^2}

    {\pi}r^3=V={\pi}r^2h

    {\pi}r^2(r-h)=0

    r=0 corresponds to the jar being unrealistically stretched
    into an "invisible" line.

    h=r

    \frac{h}{2r}=\frac{h}{2h}=\frac{1}{2}
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