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Math Help - optimization - cylinder in cone

  1. #1
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    Exclamation optimization - cylinder in cone

    Find the altitude of the cylinder with the largest volume that can be inscribed inside a right circular cone.
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  2. #2
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    See attached picture.

    H is the height of the cone; h is the height of the cylinder. R is the radius of the cone; r is the radius of the cylinder. (sorry I don't know why but I called it B and b in the diagram... just pretend B is R and b is r).

    By similar triangles,

    \frac{H}{R} = \frac{H-h}{r}

    Solve for h (it will be in terms of r).

    Plug this equation for h into the volume function for the cylinder (  V = \pi*r^2*h).

    Maximise V over [0,R].
    Attached Thumbnails Attached Thumbnails optimization - cylinder in cone-triangle.jpg  
    Last edited by MollyMillions; February 5th 2010 at 01:13 PM. Reason: Typo.
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