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Math Help - Volum of a cylinder in a cone (driving me insane!)

  1. #1
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    Volum of a cylinder in a cone (driving me insane!)

    a cylinder of radius r and height h is inscribed within a cone with bas radius 6 and height 20

    1) show that the volume of the cylinder is given by

    V= 10.(pi).r^2.(6-r)
    --
    3

    2) find r and h

    3) what is the maximum volume
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  2. #2
    MHF Contributor Unknown008's Avatar
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    First, what is the relationship between the height of the cylinder and the radius of the cylinder?

    If you make a sketch, you'll see that for a certain radius r, the cone would have a height of l = (r/6)*20, using similar triangles.



    From here, the height of the cylinder will become h = 20 - (20r)/6.

    Then, the formula for the volume of a cylinder is

    V = \pi r^2 h

    Using the equation you got, you now get:

    V = \pi r^2 (20 - \frac{20r}{6})

    Simplifying;

    V = \frac{10}{3}\pi r^2 (6-r)

    For the second part, are you sure that's the question? Because r and h would vary, depending on the volume. If the question was in fact, "Find the value of r and h, provided that the volume of the cylinder is maximised." there there would be a solution.
    Last edited by Unknown008; August 14th 2010 at 08:47 PM.
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  3. #3
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    thanks. it says to find h and r when they are maximum values so i think i shouldve posted this in calculus
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  4. #4
    MHF Contributor Unknown008's Avatar
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    Yes, you need to find the derivative of V, and solve for V' at V' = 0.

    You will get the radius in this situation, and you can then find the height using r.
    Last edited by Unknown008; August 14th 2010 at 09:47 PM.
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  5. #5
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    ok thanks
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