2004 AMC 12B Problems/Problem 19
From AoPSWiki
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
Consider a trapezoidal (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,
See also
| 2004 AMC 12B (Problems • Resources) | ||
| Preceded by Problem 18 | Followed by Problem 20 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
