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Math Help - optimization problem(minimum total area, given the volume)

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    Talking optimization problem(minimum total area, given the volume)

    A closed Cylinder container needs to have a volume of 128pie dm3. Find the dimensions if the total area must be a minimum.
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    Quote Originally Posted by dcfi6052 View Post
    A closed Cylinder container needs to have a volume of 128pie dm3. Find the dimensions if the total area must be a minimum.
    The surface area is,
    2\pi r^2+2\pi rh
    The volume is,
    \pi r^2 h=128\pi
    Thus,
    h=\frac{128}{r^2}
    Substitute that into first equation,
    2\pi r^2+2\pi r\left( \frac{128}{r^2} \right)
    Thus,
    2\pi r^2+\frac{256 \pi}{r}
    In these problems it is safe to assume that the solution appears at the critical point.
    Thus, derivative,
    4\pi r-256 \pi r^{-2}=0
    Divide by 4\pi,
    1-64 r^{-2}=0
    Solve for positive root,
    r=8
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    Quote Originally Posted by dcfi6052 View Post
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