2004 AIME II Problems/Problem 11
Problem
A right circular cone has a base with radius and height A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is , and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is Find the least distance that the fly could have crawled.
Solution
The easiest way is to unwrap the cone into a circular sector. Center the sector at the origin with one radius on the positive -axis and the angle going counterclockwise. The circumference of the base is . The sector's radius (cone's sweep) is . Setting .
If the starting point is on the positive -axis at then we can take the end point on 's bisector at radians along the line in the second quadrant. Using the distance from the vertex puts at . Thus the shortest distance for the fly to travel is along segment in the sector, which gives a distance .
See also
2004 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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