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Math Help - What is volume of cylinder ?

  1. #1
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    What is volume of cylinder ?

    Cylinder is in the sphere. Radius of sphere is r. What is the biggest volume of cylinder then ?
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  2. #2
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    Hello, totalnewbie!

    Cylinder is inscribed in a sphere of radius R.
    What is the biggest volume of cylinder then?
    Code:
                  * * *
              * - - - - - *
            * |           | *
           *  |           |  *
              |           |
          *   |           |   *
          *   |     *     |   *
          *   |     : \   |   *
              |    h:  \R |
           *  |     :   \ |  *
            * |     :  r \| *
              * - - + - - *
                  * * *

    The radius of the cylinder is r.
    The height of the cylinder is 2h.

    The volume of the cylinder is: . V \:=\:\pi r^2(2h) \:=\:2\pi r^2h [1]

    In the diagram, we see that: . r^2 + h^2\:=\:R^2\quad\Rightarrow\quad h \:=\:\sqrt{R^2 - r^2} [2]

    Substitute [2] into [1]: . V \:=\:2\pi r^2\sqrt{R^2-r^2}


    Therefore, we must maximize: . V \;= \;2\pi r^2\left(R^2-r^2\right)^{\frac{1}{2}}

    . . Go for it!

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

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

    r^{2}=R^{2}-\frac{h^{2}}{4}

    So, V={\pi}\left(R^{2}-\frac{h^{2}}{4}\right)h

    = {\pi}\left(R^{2}h-\frac{h^{3}}{4}\right)

    for 0\leq{h}\leq{2R}.

    \frac{dV}{dh}={\pi}\left(R^{2}-\frac{3}{4}h^{2}\right)

    \frac{dV}{dh}=0, when h=\frac{2R}{\sqrt{3}}.

    Volume is largest when h=\frac{2R}{\sqrt{3}} and r=\sqrt{\frac{2}{3}}R
    Last edited by galactus; November 24th 2008 at 05:39 AM.
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  4. #4
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    And then area of Cylinder side is
    \frac{4*3.14*2^{\frac{1}{2}}R^2}{3}
    Am I right ?
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