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Math Help - Applications of Differentiation

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    Applications of Differentiation

    A cylinder of volume V is to be cut from a solid sphere of radius R. Obtain its maximum volume in terms of R.

    Thanks in advance.
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  2. #2
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    Let r and h be the radius and height of the cylinder.

    The cylinder volume is V={\pi}r^{2}h

    But, by Pythagoras, r^{2}+\left(\frac{h}{2}\right)^{2}=R^{2}

    Thus, 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)

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

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

    If h=0, \;\ \frac{2R}{\sqrt{3}}, \;\ 2R, then

    V=0, \;\ \frac{4\pi}{3\sqrt{3}}R^{3}, \;\ 0

    so the volume is the largest when h=\frac{2R}{\sqrt{3}}, \;\ r=\sqrt{\frac{2}{3}}R
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