1990 AMC 12 Problem 25

Question

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?

1-\frac{\sqrt3}{2}

\frac{2\sqrt3-3}{2}

\frac{\sqrt2}{6}

\frac14

\frac{\sqrt3(2-\sqrt2)}4

Answer: B

Concepts:three-dimensional geometry distance formula tangent spheres

Difficulty rating: 2260

Small hint:

Place one corner sphere’s center at

(r,r,r)

Big hint:

Its distance from

(\frac{1}{2},\frac{1}{2},\frac{1}{2})

equals

2r

Solution:

If the radius is

r,

a corner sphere has center

(r,r,r)

and the central sphere has center

(\frac{1}{2},\frac{1}{2},\frac{1}{2}).

Tangency gives

\sqrt3\left(\frac12-r\right)=2r.

Solving and rationalizing,

r=\frac{\sqrt3}{2(2+\sqrt3)} =\frac{2\sqrt3-3}{2}.

Thus the correct answer is B.