2014 AMC 12A Problem 17

Problem 17 of 25IntermediateGeometry

4\times4\times h

rectangular box contains a sphere of radius

2

and eight smaller spheres of radius

1.

The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is

h?

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Solution

Place the box with a corner at the origin. Each small sphere sits in a corner with center

1

unit from three faces. The four top small-sphere centers form a square of side

2,

whose center lies on the box axis; a corner of that square is

\sqrt2

from the center. The big sphere’s center is on the axis, at distance

2+1=3

from each top small center. The vertical gap between them is

\sqrt{3^2-\sqrt2^2}=\sqrt7.

The big center is at height

\dfrac h2

and the top small centers at height

h-1,

(h-1)-\dfrac h2=\sqrt7,

giving

\dfrac h2=1+\sqrt7

and

h=2+2\sqrt7.

Thus, the correct answer is A.