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Math Help - Sphere, Geometry Problem

  1. #1
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    Sphere, Geometry Problem

    A cone circumscribes a sphere and has its slant height equal to the diameter of its base.
    Show that the volume of the cone is 9/4 the volume of the sphere.

    Basically I need to show my professor that the volume of the cone is 9/4 the volume of the sphere.
    Can someone help me with it or just give me like 3 solution steps so I can proceed. Thanks in advance.
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  2. #2
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    Re: Sphere, Geometry Problem

    If youcut the cone in two halves, you will get an equilateral triangle its vertices are vertices of an hexagon and its height is 3R/2 and R is the radius of the sphere. Continue from there
    Thanks from PhillipMasagca
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  3. #3
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    Re: Sphere, Geometry Problem

    cone has an equilateral cross section side length d
    altitude ofcone =1/2drad3
    radius ofsphere=1/2 d/rad3
    volume of cone /volume sphere wiil cancel out to 9/4
    Thanks from PhillipMasagca
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  4. #4
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    Re: Sphere, Geometry Problem

    I'm still confused. Is the radius of the sphere equal to the radius of the cone?
    Thanks for replying i appreciate your feedbacks
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  5. #5
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by PhillipMasagca View Post
    I'm still confused. Is the radius of the sphere equal to the radius of the cone?
    Thanks for replying i appreciate your feedbacks
    No it is not. Use the information in the attached and use the properties of an equilateral triangle.
    Sphere, Geometry Problem-untitled2.gif
    Thanks from PhillipMasagca
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  6. #6
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by PhillipMasagca View Post
    I'm still confused. Is the radius of the sphere equal to the radius of the cone?
    Thanks for replying i appreciate your feedbacks
    look closely at the values I gave you for base radius and sphere radius and note they are not equal to each other
    Thanks from PhillipMasagca
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  7. #7
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by bjhopper View Post
    look closely at the values I gave you for base radius and sphere radius and note they are not equal to each other
    My apology. the center of an inscribed circle is the intersection point of the bisectors. In an equilateral triangle a bisector is also a median. The intersection point of the bisectors is also the intersection point of the medians. Therefore, in your problem the radius of the inscribed circle is 1/3 the length of the median, which is also the height of the equilateral triangle. You need to correlate the side of the triangle with the radius of the circle via the length of the height.

    Sphere, Geometry Problem-untitled2.gif
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  8. #8
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    Re: Sphere, Geometry Problem

    @bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

    @votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)
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  9. #9
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by PhillipMasagca View Post
    @bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

    @votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)
    I decipher 1/2drad3 as (1/2)*diameter*radical(3), radical here means sqrt

    You got the radius of the sphere correctly assuming the height of the cone is one unit. but if you label the height of the cone h units, What the radius of the sphere would be.

    Now you will need to relate h to the side of the equilateral triangle, which is the base of the triangle (why?)
    Last edited by votan; September 24th 2013 at 05:10 AM. Reason: adding note
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  10. #10
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by PhillipMasagca View Post
    @bjhopper what is 1/2drad3? is it 1/2(diameter)(radius^3)?? (Thanks Again)

    @votan after i get the radius of the sphere to be 1/3 of the height of the cone what is the next step. im kinda confused (Thanks Again)

    1/2 *d*rad3 = alt of cone
    1/2 *d /rad3 = radius of sphere
    Both are properties of 30-60-90 triangles
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  11. #11
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by bjhopper View Post
    1/2 *d*rad3 = alt of cone
    1/2 *d /rad3 = radius of sphere
    Both are properties of 30-60-90 triangles
    Radius of sphere is third altitude of cone. The sphere is inscribed in the cone.
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  12. #12
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    Re: Sphere, Geometry Problem

    Quote Originally Posted by votan View Post
    Radius of sphere is third altitude of cone. The sphere is inscribed in the cone.

    1/2*d*1/rad3 = radius of sphere = 1/3 of alt of cone
    1/2*d*1/rad 3=1/2*d*rad3/rad3^2=1/6*d*rad3=1/3*d*1/2*rad3
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  13. #13
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    Re: Sphere, Geometry Problem

    Sphere, Geometry Problem-untitled2.gif
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