Problem:
A 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 ?
Answer Choices:
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Solution:
Connect the centers of the large sphere and the four small spheres at the top to form an inverted square pyramid as shown. Since is a square of side . Also, and is a right triangle, so
The distance from the plane containing to the top of the box is 1 . Therefore the total height of the box is .
The problems on this page are the property of the MAA's American Mathematics Competitions