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Escaping the Tiger

Date: 07/10/2003 at 16:42:44
From: Peter
Subject: Circles, speed, and distance

A man stands in the center of a circle. On the circumference of the 
circle is a tiger that is constrained to move only around the circle. 
The tiger can run four times as fast as the man. How can the man 
escape the circle without being eaten by the tiger? (The tiger will 
try to minimize its distance from the man at all times).


Date: 07/17/2003 at 08:13:41
From: Doctor George
Subject: Re: Circles, speed, and distance

Hi Peter,

I've talked over your problem with one of the other math doctors, 
and here is the easiest way we have come up with to show that the 
man can escape from the lion. Here are the important points.

1. The best course of action for the man is to get as far from the 
   lion as possible, exactly on the opposite side of the center from 
   the lion.

2. For a radius R, as long as the man is less that R/4 from the 
   center he can move around the center faster than the lion.

3. Since the lion cannot stop the man from getting to a position R/4 
   on the other side of the center, there is no point in the lion 
   wasting energy until the man is that far away.

4. If the lion just stays put, the man can start by backing straight 
   away from the lion until he is R/4 away from the center.

5. Now the man can make a break for the oppposite side and get there 
   before the lion.

The math only really gets interesting if the lion decides to waste 
some energy to keep the man from backing directly away from him. If 
the lion starts at the north side of the circle and runs counter- 
clockwise, the man can stay on the opposite side of the center by 
running on the southern semi-circle connecting (0,0) and (R/4,0) 
until the lion is due west of center.

Here is a similar problem if you care to dig deeper.

   Ponder This - May 2001 Challenge, IBM
   (Martin Gardner - escaping the goblin)
   http://domino.research.ibm.com/Comm/wwwr_ponder.nsf/challenges/
May2001.html

Write again if you have more interest in this.

- Doctor George, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Conic Sections/Circles
Middle School Conic Sections/Circles
Middle School Word Problems

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