1999 AHSME Problems/Problem 29
Problem
A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point is selected at random inside the circumscribed sphere. The probability that lies inside one of the five small spheres is closest to
Solution
Let the radius of the large sphere be , and of the inner sphere . Label the vertices of the tetrahedron , and let be the center. Then pyramid , where denotes volume; thus . Since and are both pyramids that share a common face , the ratio of their volumes is the ratio of their altitudes to face , so . However, , so it follows that . Then the radius of an external sphere is .
Since the five described spheres are non-intersecting, it follows that the ratio of the volumes of the spheres is .
See also
1999 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 28 |
Followed by Problem 30 | |
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