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Thread: Optimization; cylinder inside sphere

  1. #1
    Junior Member Tclack's Avatar
    Oct 2009
    No permanant location.

    Optimization; cylinder inside sphere

    Find the dimensions(r and h) of the right circular cylinder of greatest Surface Area that can be inscribed in a sphere of radius R.

    my work so far

    $\displaystyle SA=2\pi r^2+2\pi rh $

    $\displaystyle r^2 + (\frac{h}{2})^2 = R^2$

    $\displaystyle h=2\sqrt{R^2-r^2}$

    $\displaystyle SA=2\pi r^2+4\pi r\sqrt{R^2-r^2} $

    $\displaystyle \frac{dSA}{dr}=4\pi r+4\pi (\sqrt{R^2-r^2}+\frac{-2r^2}{2\sqrt{R^2-r^2}})$

    I tried setting that equal to zero, but I wasn't coming up with the right answer

    The answer in the book(not mine): $\displaystyle r=\sqrt{\frac{5+\sqrt{5}}{10}}R$
    $\displaystyle h=2\sqrt{\frac{5-\sqrt{5}}{10}}R$

    Can anyone see my error, or did I make one?
    Last edited by Tclack; Oct 13th 2009 at 03:54 PM. Reason: clarification
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