1999 AMC 12 Problem 29
Attempt Problem 29 of the 1999 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1999 AMC 12 solutions, or check the answer key.
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29.
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
Answer: C
Solution:
Let be the common center of the inscribed and circumscribed spheres. Splitting the tetrahedron into four congruent pieces from shows the circumradius is times the inradius, so the circumscribed sphere has times the inscribed sphere's volume
If one of the four other small spheres has radius its center is from and also from so These five spheres have disjoint interiors (the central sphere is tangent to each of the other four), so their union has volume The probability is closest to
Thus, the correct answer is C.