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 <img alt="$P$" height="12" src="https://latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png" width="14" /> is selected at random inside the circumscribed sphere. The probability that <img alt="$P$" height="12" src="https://latex.artofproblemsolving.com/4/b/4/4b4cade9ca8a2c8311fafcf040bc5b15ca507f52.png" width="14" /> lies inside one of the five small spheres is closest to   <img alt="$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4$" height="18" src="https://latex.artofproblemsolving.com/d/4/f/d4f227bb4893c3c158c852ad4ac78d5b061160c2.png" width="399" />

Question No: 317

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 $P$ is selected at random inside the circumscribed sphere. The probability that $P$ lies inside one of the five small spheres is closest to

$\mathrm{(A) \ }0 \qquad \mathrm{(B) \ }0.1 \qquad \mathrm{(C) \ }0.2 \qquad \mathrm{(D) \ }0.3 \qquad \mathrm{(E) \ }0.4$

Editorial

Tags: Geometry Probability

Question Details