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Math Help - twist on an old optimization problem

  1. #1
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    twist on an old optimization problem

    We've all seen the classic 'sphere inside a cone', cylinder in a cone', cone in a sphere' problems and etc.

    Here's one that is a little different if anyone would enjoy tackling it.

    "Find the ellipsoid of max volume which can be inscribed in a cone of radius R and height H".

    Any willing participants?. The old, cliche, max min and related rates problem have become rather worn out for most of us, I would assume. Though, they are still challenging for calc students. I just thought I would add a twist. Something a little different.
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  2. #2
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    Hello, galactus!

    I think I've got a handle on this one . . .


    We've all seen the classic 'sphere inside a cone', 'cylinder in a cone',
    'cone in a sphere' problems, etc.

    Here's one that is a little different if anyone would enjoy tackling it.

    \text{Find the ellipsoid of max volume which can be inscribed in a cone of radius }R \text{ and height }H

    Let's orient the ellipsoid 'horizontally' . . . then we have:
    Code:
        -           *
        :          /|\
        :         / | \
        :        /  |  \
        :       /   |   \
        H      /  ..*..  \
        :     /*::::|::::*\
        :    *-:-:-:+ - a -*
        :   /  *::::b::::*  \
        -  *- - - - * - - - -*
                    :    R   :

    From similar right triangles, we have: . \frac{a}{H-b} \:=\:\frac{R}{H}\quad\Rightarrow\quad b \:=\:\frac{H}{R}(R-a)\;\;{\color{blue}[1]}

    The volume of this ellipsoid of revolution is: . V \;=\;\frac{4}{3}\pi a^2b\;\;{\color{blue}[2]}

    Substitute [1] into [2]: . V \;=\;\frac{4\pi}{3} a^2\cdot\frac{H}{R}(R-a) \;=\;\frac{4\pi H}{3R}(Ra^2-a^3)

    Maximize: . V' \;=\;\frac{4\pi H}{3R}(2Ra - 3a^2) \:=\:0\quad\Rightarrow\quad a(2R -3a) \:=\:0\quad\Rightarrow\quad\boxed{ a \:=\:\frac{2}{3}R}

    Substitute into [2]: . b \:=\:\frac{H}{R}\left(R - \frac{2}{3}R\right)\quad\Rightarrow\quad\boxed{ b \:=\:\frac{1}{3}H}

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  3. #3
    Eater of Worlds
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    Cool, Soroban.
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