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Math Help - Sphere inside a cone problem.

  1. #1
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    Question Sphere inside a cone problem.

    I'm taking a test soon, the next problem is one of which I'll be facing on it, and I would like to ask you guys to help me know how to solve it, as well as which is what I need to study in order to solve alike problems.

    Suppose that a sphere of radius 1 is inscribed in a right circular cone with radius r and height h.

    (1) Express r in terms of h.




    (2) Find the minimum of the volume of such a right circular cone.

    ---------------------------------------------------------



    Now the answer for (1) is supposed to be


    \sqrt{(h-1)^2 - 1} : 1 = h:r

    h = r\sqrt{h^2 - 2h}

    r = \frac{h}{\sqrt{h^2 - 2h}}

    Why is this so?

    -------------------------------------------------------

    And for (2)

    <br />
V (h) = \frac{1}{3}\pi r^2h = \frac{1}{3} \pi h \cdot \frac{h^2}{h^2 - 2h} = \frac{1}{3} \pi \cdot \frac{h^2}{h - 2}

    V'(h) = \frac{1}{3}\pi \cdot \frac{h (h - 4)}{(h - 2)^2}

    \frac{8\pi}{3}



    Thanks.
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  2. #2
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    “Why is this so?”
    Because triangle AEO is similar to triangle ADC.
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  3. #3
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    Thanks, I think I'm not yet prepared to take this kind of test, I think I'll keep studying and try it in December, but thanks for your help
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