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Math Help - maximum volume

  1. #1
    Junior Member ihmth's Avatar
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    maximum volume

    Find the circular cone of maximum volume inscribed in a sphere of radius a.


    I can visualize what the problem is trying to say, but the problem is..I don't know the solution Help

    -thank you
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  2. #2
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    earboth's Avatar
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    Quote Originally Posted by ihmth View Post
    Find the circular cone of maximum volume inscribed in a sphere of radius a.


    I can visualize what the problem is trying to say, but the problem is..I don't know the solution Help

    -thank you
    I've attached a sketch of the situation.

    1. You are dealing with a right triangle. According to Euklids theorem you have:

    r^2=h\cdot p

    and : h = 2a - p~\implies~p=2a-h

    2. The volume of the cone is calculated by:

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

    3. Substitute :

    V_{cone} = \frac13 \pi \cdot h\cdot p \cdot h and

    V_{cone} = \frac13 \pi \cdot h\cdot (2a-h) \cdot h

    4. You have now the volume as a function of h:

    V(h)=-\frac13\pi\cdot h^3 + \frac23 \pi\cdot h^2

    5. Calculate the first derivation of V(h) and solve V'(h) = 0 for h.

    I've got h = \frac43 a
    Attached Thumbnails Attached Thumbnails maximum volume-kegel_inkugel.gif  
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