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Show that a right-circular cylinder of greatest volume that can be inscribed in a right-circular cone that has a volume that is 4/9 the volume of the cone.

  • mathe -

    Let the cone radius = R
    cone height = H
    Let cylinder height = h
    then cylinder radius, r = R(1-h/H)

    Volume of cylinder, V

    For maximum volume,
    which simplifies to:
    which when solved for h gives
    h=H/3 or h=H
    h=H will give a zero volume (min.) so reject.
    for h=H/3,
    Volume of cylinder
    =4/9 volume of the cone.

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