All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Six Bugs (Posted on 2004-03-16) Difficulty: 4 of 5
Remember "Four Bugs" or "Three Bugs"?

In this problem, the six bugs start at the corners of a regular hexagon (with side length=10 inches).

Again, the bugs travel directly towards their neighbor (counter-clockwise). And, again, each bug homes in on its target, regardless of its target's motion. So, their paths will be curves spiraling toward the center of the hexagon, where they will meet.

What distance will the bugs have covered by then, and how did you determine it?

See The Solution Submitted by SilverKnight    
Rating: 3.2500 (4 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Problem Solution With Explanation | Comment 7 of 8 |
(In reply to answer by K Sengupta)

Let the regular hexagon be denoted by ABCDEF. At any given instant, the three bugs located at each of the vertices A, B, C, D, E and F of the hexagon which shrinks and rotates as the six  bugs move closer together. Thus, the six  bugs will describe six congruent logarithmic spirals that will meet at the center of the hexagon.

Since each angle of a regular hexagon is 120 degrees, it follows
that the path of the pursuer and the pursued will describe an
angle of 120 degrees between them and consequently, each bug’s motion will have a component equal to cos 120 = - 0.5 times its velocity that will carry it towards its pursuer. Thus, these two bugs will have a mutual approach speed of  1 - 0.5 = 0.5 times the velocity of each bug.

Consequently, the distance traversed by each of the six  bugs
until they meet at the center of the regular hexagon triangle
= 10/(0.5)
= 20 inches


  Posted by K Sengupta on 2007-05-16 05:47:45
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (2)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information