2004 AMC 12B Problems/Problem 19
Problem
A truncated cone has horizontal bases with radii and . A sphere is tangent to the top, bottom, and lateral surface of the truncated cone. What is the radius of the sphere?
Solution
Solution 1
Consider a trapezoid (label it as follows) cross-section of the truncate cone along a diameter of the bases:
Above, and are points of tangency. By the Two Tangent Theorem, and , so . We draw such that it is the foot of the altitude to :
By the Pythagorean Theorem,
Solution 2
Create a trapezoid with inscribed circle exactly like in Solution #1, and extend lines and from the solution above and label the point at where they meet . Because = , = . Let and .
Because these are radii, . so . Plugging in, we get so .Triangles and are similar so which gives us . Solving for x, we get and
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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All AMC 12 Problems and Solutions |
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