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Math Help - Maximum volume of cone in sphere

  1. #1
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    Maximum volume of cone in sphere

    What is the maximum volume of a cone which fits inside a sphere of volume 1?
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  2. #2
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    Quote Originally Posted by Suhada View Post
    What is the maximum volume of a cone which fits inside a sphere of volume 1?
    1. Draw a sketch. Preferable a sketch of the vertical cross-section.

    2. From the volume you can determine the radius of the sphere:

    V_{sphere} = 1 = \frac43 \cdot \pi \cdot a^3

    Solve for a.

    3. Let r denote the radius of the base circle of the cone and h it's height.

    Then

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

    4. According to Euclid's theorem

    (2a-h) h = r^2

    5. Plug in this term into the equation of the volume to get the equation of a function:

    V(h)= \frac13 \cdot \pi \cdot (2a-h) h \cdot h = \frac13 \cdot \pi \cdot (2ah^2-h^3)

    6. Determine the maximum of this function.
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  3. #3
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    Thanks earboth.
    I have solved this problem before but thought that some of you might enjoy it.

    I really like this problem because it's so simple to state and yet involves quite a range of mathematical techniques form algebra, geometry & calculus.

    Anyone care to take it to a conclusion?

    Suhada
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