Problem:
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let be the radius of each sphere. Note that the centers of the eight outer spheres form a cube of side whose main diagonal is units. Since the length of the diagonal of a cube is times its side, . Solve this equation to find .
If the radius of each sphere is , the center of a corner sphere is units from the closest vertex. Thus the length of the diagonal of the cube is . But the length of the diagonal of a unit cube is . Solve to find .
Note. To visualize the arrangement of the spheres in the cube, begin with nine small congruent spheres with one at the center and one tangent to the three faces at each of the eight vertices of the cube. Keeping the center sphere in the center of the cube and the other eight tangent to their three faces, expand the radii of all nine spheres until the spheres are tangent.