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 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.
![\[AB = \sqrt{(375\sqrt{2})^2 + 125^2 - 2 \cdot (375\sqrt{2})(125)(\cos 135^\circ)} = \boxed{625}\]](http://latex.artofproblemsolving.com/0/7/8/0781c1bc8c553c441595491e9f8901c8fdd15a15.png)
~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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