perplexus.info :: Just Math : Fly on a cone

Fly on a cone (Posted on 2009-07-01)

A right circular cone has radius r and slant height s.
A fly starts at a point on the edge of the cone's base,
walks around the vertex, and returns to its starting point.

What is the minimum distance traveled in terms of r and s?

Submitted by Bractals

Rating: 4.0000 (1 votes)

Solution: (Hide)

Cut the cone along the the line from the starting point to the vertex. Unfold the cone flat into a sector of a disk with center C and radius s. Label the starting and ending points as A and A' respectively. The arc length AA' is 2πr and the measure of angle ACA' is 2πr/s radians. If m(/ACA') ≥ π ( 2r ≥ s ), then the fly walks along the cut from A to C and then from C to A' for a distance of 2s else the fly walks the straight line AA' for an easily calculated distance of 2s*sin(πr/s)

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-02-06 23:18:37
solution	Charlie	2009-07-01 13:46:23

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