2014 AMC 12A Problems/Problem 17
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 ?
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 .
Let be the center of the large sphere and be the center of any small sphere. Let be a vertex of the rectangular prism closest to point . Let be the point on the edge of the prism such that and are perpendicular. Let points and point lie on and respectively such that and are perpendicular at .
is the radii of the spheres, so . is the shortest length between the center of a small sphere and the edge of the prism, so . Similarly, . Since is a rectangle, . Since , . Then, . is the length from to the top of the prism or . Thus, . The prism is symmetrical, so
Take a cross section and you will see that h is made up of the two radii of the circles plus some radical expression. The only choice satisfying this condition is .
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