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Math Help - Problem 23

  1. #1
    Grand Panjandrum
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    Problem 23

    Consider a tunnel (not necessarily straight) of length 101 units through a
    sphere of radius 51 units.

    Prove that the sphere can be cut into two congruent pieces one of which
    contains the tunnel in its interior.

    RonL
    Last edited by CaptainBlack; May 28th 2007 at 08:14 AM.
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  2. #2
    Senior Member ecMathGeek's Avatar
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    Quote Originally Posted by CaptainBlack View Post
    Consider a tunnel (not necessarily straight) of length 101 units through a
    sphere of radius 51 units.

    Prove that the sphere can be cut into two congruent pieces one of which
    contains the tunnel in its interior.

    RonL
    How can these two pieces be congruent if one has a tunnel in it?
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by ecMathGeek View Post
    How can these two pieces be congruent if one has a tunnel in it?
    The two pieces of the sphere ignoring the hole are congruent

    RonL
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  4. #4
    Grand Panjandrum
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    Solution:

    Let the entry and exit points be A and B. Then all posible paths of length
    101 lie in an ellipsoid such that the sum of the distance from A and that from
    B is less than 101 units. But the centre of the sphere cannot be inside this
    ellipsoid as the sum of the distance from A to the centre and from B to the
    centre is 102.

    Now the plane through the centre normal to the plane containing the points
    A, B and the centre does not intersect the ellipsoid, and so divides the sphere
    (ignoring the hole) into two equal hemisphere.

    RonL
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