2004 AIME II Problems/Problem 11
Contents
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 1
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
.
Solution 2
To find the shortest length from the red to blue points, the net of the side of the cone could be drawn.
The angle is equal to
, or
. Therefore, the law of cosines could be utilized.
~Diagram and Solution by MaPhyCom
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 | ||
All AIME Problems and Solutions |
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