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Math Help - Sphere in Cone Geometry

  1. #1
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    Sphere in Cone Geometry




    That's a question on my textbook.... i can't solve it and my teacher is away. Anyone help?
    Thanks.
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  2. #2
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    Quote Originally Posted by anonymous_maths View Post
    ...
    That's a question on my textbook.... i can't solve it and my teacher is away. Anyone help?
    Thanks.
    1. The cross-section of the cone is an equilateral triangle with AM = R.
    In the top right triangle you get:

    R \cdot \tan(30^\circ) = r~\implies~R=\dfrac r{\tan(30^\circ)} = \dfrac r{\frac13 \sqrt{3}}~\implies~\boxed{R=r \cdot \sqrt{3}}

    2. Let denote x the distance in question. Use Pythagorean theorem:

    R^2+r^2=(r+x)^2~\implies~3r^2+r^2 = (r+x)^2~\implies~2r=r+x~\implies~x=r

    Unfortunately I have to leave now ...
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  3. #3
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    Quote Originally Posted by anonymous_maths View Post
    ...
    That's a question on my textbook.... i can't solve it and my teacher is away. Anyone help?
    Thanks.
    Let h denote the height of the cone. Then:

    h^2+R^2=(2R^2)~\implies~h=R\cdot \sqrt{3}

    The volume of a cone is calculated by:

    V=\dfrac13 \cdot \pi \cdot R^2 \cdot h

    That means:

    V=\dfrac13 \cdot \pi \cdot R^2 \cdot R \cdot \sqrt{3} = \dfrac{\pi \sqrt{3}}3 R^3
    ------------------------------------------------------------------------------------
    From a) you know that
    R=r \cdot \sqrt{3}~\implies~r=\dfrac{\sqrt{3}}3 R

    The volume of the sphere is calculated by:

    V=\dfrac43 \pi r^3

    That means:

    V=\dfrac43 \pi \left(\dfrac{\sqrt{3}}3 R\right)^3 = \dfrac{4 \pi  \sqrt{3}}{27} R^3
    ------------------------------------------------------------------------------------
    The percentage is calculated by percentage = \dfrac{partial\ amount}{total\ amount} \cdot 100

    p=\dfrac{\dfrac{4\pi \sqrt{3}}{27} R^3}{\dfrac{\pi \sqrt{3}}3 R^3}\cdot 100 = \dfrac49 \cdot 100 \approx 44.4\%
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