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Math Help - spheres

  1. #1
    Member Veronica1999's Avatar
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    spheres

    Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

    The answer is 3+ root 69/3.
    I really can't think of an approach, yet...
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  2. #2
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    The larger sphere would never be able to rest on two smaller spheres without falling... This question = FAIL!
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  3. #3
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    Quote Originally Posted by Veronica1999 View Post
    Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?

    The answer is 3+ root 69/3.
    I really can't think of an approach, yet...
    1. The centers (C_1, C_2, C_3)of the smaller spheres form a isosceles triangle with the side-length 2.

    2. The center M of the larger sphere is vertically above the centroid F of the isosceles triangle. The larger sphere is tangent to all three smaller spheres.

    3. MFC_1 is a right triangle with
    hypotenuse |\overline{MC_1}| = 3
    horizontal leg |\overline{FC_1}| = \frac 23 \cdot \sqrt{2^2-1^2}=\frac 23 \cdot \sqrt{3}
    vertical leg |\overline{MF}| =\sqrt{3^2-\left(\frac 23 \cdot \sqrt{3}  \right)^2} = \sqrt{\frac{23}{3}}

    4. You have to add one small radius and one large radius to the distance |\overline{MF}| and you'll get the given result.

    5. For additional information have a look here: Close-packing of spheres - Wikipedia, the free encyclopedia
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  4. #4
    Senior Member abhishekkgp's Avatar
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    Quote Originally Posted by Prove It View Post
    The larger sphere would never be able to rest on two smaller spheres without falling... This question = FAIL!
    some glue could be used.
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  5. #5
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    Quote Originally Posted by abhishekkgp View Post
    some glue could be used.
    It would still be top heavy and still fall...
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  6. #6
    Member Veronica1999's Avatar
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    Thank you so much Earboth!
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  7. #7
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    Re: spheres

    also see Earboths solution of a four ball pile 12/22.10
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  8. #8
    Member Veronica1999's Avatar
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    Re: spheres

    Thanks for the info. I really appreciate it.
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