Problem:
A truncated cone has horizontal bases with radii 18 and 2. A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
Answer Choices:
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Solution:
Let and be parallel diameters of the bottom and top bases, respectively. A great circle of the sphere is tangent to all four sides of trapezoid . Let , and be the points of tangency on , and , respectively. Then
so . If is on such that is a right angle, then . Thus
and the radius of the sphere is .
The problems on this page are the property of the MAA's American Mathematics Competitions