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:
Use the Pythagorean Theorem to conclude that the distance from the vertex of the cone to a point on the circumference of the base is
Cut the cone along the line through the vertex of the cone and the starting point of the fly, and flatten the resulting figure into a sector of a circle with radius . Because the circumference of this circle is and the length of the sector's arc is , the measure of the sector's central angle is . The angle determined by the radius of the sector on which the fly starts and the radius on which the fly stops is . Use the Law of Cosines to conclude that the least distance the fly could have crawled between the start and end positions is
which can be simplified to .
The problems on this page are the property of the MAA's American Mathematics Competitions