Minimum Distance

Date: 22 May 1995 22:29:15 -0400
From: Lew England
Subject: Finding Minimum Distance

To whom it may concern:

I am preparing to give a multimedia presentation concerning the Internet
and Education to a large group of teachers, students, and parents.

I hoped you could answer the following question so I could provide an
example when I refer students to your program. This was taken from a school
geometry book:

"Inside a rectangular room, measuring 30' in length and 12' in width and
height, a spider is at a point on the middle of one of the end walls, 1
foot from the ceiling, as shown at A; and a fly is on the opposite wall, 1
foot from the floor in the center, as shown at B. What is the shortest
distance that the spider must crawl in order to reach the fly, which
remains stationary? Of course the spider never drops or uses its web, but
crawls fairly."

Thank you for your consideration.

Sincerely,
Bob England

P.S. I have also provided an example in GIF format.

   

Date: 23 May 1995 00:45:08 -0400
From: Dr. Ken
Subject: Finding Minimum Distance

Hello there!

I'm glad to hear about your upcoming presentation, and I hope it goes well. 

The trick to this problem is realizing that you can cut the spider's box
open and lay it flat.  Then the problem becomes an exercise in good old
plane geometry.  Here are three different ways you could lay the box out,
where the x is the spider and the o is the fly:

      12                30                 12
  -----------------------------------------------
  |       |                             |       |
  |       |                             |       |
  |      x|                             |      o| 12
  |       |                             |       |
  |       |                             |       |
  -----------------------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------


      12                30               
  ---------------------------------------
  |       |                             |       
  |       |                             |       
  |      x|                             |       
  |       |                             |       
  |       |                             |  12     
  -----------------------------------------------
          |                             |       |
          |                             |       |
          |                             |       | 12
          |                             |       |
          |                             |   o   |
          ---------------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------


      12                30               
  ---------------------------------------
  |       |                             |
  |       |                             |
  |      x|                             |
  |       |                             |
  |       |                             |
  ---------------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |   12
          ---------------------------------------
          |                             |       |
          |                             |       |
          |                             |o      | 12
          |                             |       |
          |                             |       |
          ---------------------------------------
          |                             |
          |                             |
          |                             |
          |                             |
          |                             |
          -------------------------------

Whew!  Laying the boxes out this way, it is pretty clear that our only
options for the spider's shortest paths are the ones that we make by
connecting the spider and the fly via a straight line in these diagrams.
The problem then becomes an exercise in the Pythagorean theorem to find the
three distances, and then comparing them to see which is the shortest.  I'll
let you take it from here, and if you have any questions, don't hesitate to
write back!

-K