2014 AMC 12A Problems/Problem 17
Problem
A rectangular box contains a sphere of radius and eight smaller spheres of radius . 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 ?
Solution
Let be the point in the same plane as the centers of the top spheres equidistant from said centers. Let be the analogous point for the bottom spheres, and let be the midpoint of and the center of the large sphere. Let and be the points at which line intersects the top of the box and the bottom, respectively.
Let be the center of any of the top spheres (you choose!). We have , and , so . Similarly, . and are clearly equal to the radius of the small spheres, . Thus the total height is , or .
(Solution by AwesomeToad)
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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