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Math Help - Optimization Question 2

  1. #1
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    Exclamation Optimization Question 2

    Find the volume of the largest cylinder that can be inscribed in a sphere of radius, r.
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  2. #2
    MHF Contributor chisigma's Avatar
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    In order to define the problem please observe this two-dimensional diagram…



    If we indicate with r_{c} the radius of base and with h_{c} the height of inscribed cone is…

    r_{c} = r \cdot \cos \theta

    h_{c} = 2\cdot r\cdot \sin \theta

    The volume of cylinder is therefore…


    V= 2\cdot \pi\cdot r^{3}\cdot \sin \theta \cdot \cos \theta

    Taking the derivative respect to \theta we obtain with little amount of work…


    \frac{dV}{d\theta}= 2\cdot \pi\cdot r^{3}\cdot (cos^{2} \theta -\sin^{2} \theta) (1)

    Now we have to find the value of \theta for which the derivative vanish and chose the value which maximises the volume of cone. The (1) vanish for…

    \sin^{2} \theta = \cos^{2} \theta

    … and that happens for \theta = \frac {\pi}{4}. The maximum volume of an inscribed cylinder is then…

    V= \pi \cdot r^{3}

    Very easy…

    Kind regards

    \chi \sigma
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