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Math Help - Optimization: Cone inside a larger cone

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    Optimization: Cone inside a larger cone

    A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h=1/3H.
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    Re: Optimization: Cone inside a larger cone

    Quote Originally Posted by AXQ4286 View Post
    A cone with height h is inscribed in a larger cone with height H so that its vertex is at the center of the base of the larger cone. Show that the inner cone has maximum volume when h=1/3H.
    1. Draw a sketch!

    2. Let R denote the radius of the larger cone and r the radius of the smaller cone.

    Use proportions:

    \frac rR = \frac{H-h}{H}

    Solve for r.

    3. The volume of the smaller cone is calculated by:

    V_{small\ cone}=\frac13 \cdot \pi \cdot r^2 \cdot h

    Replace r by the result of #2.

    4. Use calculus to determine the maximum value of V_{small\ cone}
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