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May 25th, 2007, 11:56 AM
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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
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Last edited by CaptainBlack; May 28th, 2007 at 10:14 AM.
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May 25th, 2007, 01:38 PM
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Quote:
Originally Posted by CaptainBlack  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? |
May 25th, 2007, 02:31 PM
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Quote:
Originally Posted by ecMathGeek 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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Truth does not change because it is, or is not, believed by a majority of the people.
Giordano Bruno |
May 28th, 2007, 10:14 AM
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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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Truth does not change because it is, or is not, believed by a majority of the people.
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