Segmenting Paths

Date: 08/20/98 at 19:23:58
From: Amber
Subject: Geometry points, lines, and planes

The question says:

   A path between opposite vertices of the square is made up of 
   hundreds of horizontal and vertical segments. What is the best
   approximation to the length of the path - 24, 34, 44, or more 
   than 44?

It has a diagram of a box with 17 on top, bottom, and both sides, with 
steps going up to the top of the box. Do you think that you can help 
me figure this problem out? I don't know how to get started.

Date: 08/21/98 at 13:17:26
From: Doctor Peterson
Subject: Re: Geometry points, lines, and planes

Hi, Amber. This is a good thinking question. One way to think about it 
is to picture a simpler case, then see how things would change if you 
had hundreds of segments.

Here's your square with just one horizontal and one vertical segment 
making the path:

           17
   +----------------+
   |                |
   |                |
   |                |
   |                | 17
   |                |
   |                |
   |                |
   +----------------+

How long is that path? (It's supposed to be right along the top and 
right edges of the square.) It should be 17 + 17, right? Now let's 
break it up into four pieces:

      6
   +------+---------+
   |      |         |
   |    10|         |
   |      |         |
   |      |    11   |
   |      +---------+
   |                |7
   |                |
   +----------------+

How has the length changed? Now it's 6 + 10 + 11 + 7. Now what happens 
if we keep on breaking it up into smaller segments?

   +---+------------+    
   |   |            |
   |   |            |
   |   +--+         |
   |      |         |
   |      +----+    |
   |           |    |
   |           |    |
   +-----------+----+

This should give you an idea how to answer the question.

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/