The Spider and the Fly

Date: 12/23/1999 at 02:52:24
From: keith
Subject: Algebra 1 honors

Hi:

I have not tried this question myself because I don't know how to 
start the problem. Here it is:

A spider and a fly are on opposite walls of a rectangular room. The 
spider asks the fly if it can come over and "visit" the fly. The fly 
believes that the shortest distance, walking on one of the room's 
surfaces, is 42 feet. (That is the distance if the spider walks 
straight up the wall, over the ceiling, and straight down the opposite 
wall.) So the fly agrees to the visit, provided the spider can find a 
path that is shorter than 42 feet. Use the "unfolded room" and the 
Distance Formula to explain the fly's fatal miscalculation.

Any help given would be greatly appreciated.

Date: 12/23/1999 at 09:41:24
From: Doctor Rob
Subject: Re: Algebra 1 honors

Thanks for writing to Ask Dr. Math, Keith.

Whether the fly lives or dies depends on the location of the spider 
and the fly on their respective walls, which you have not told in the 
above description. If both are centered horizontally between the side 
walls, the fly is D feet below the ceiling, the spider is d feet below 
the ceiling, the distance between their walls is L, the width of the 
room is W, and its height is H, then you know that L + D + d = 42, or 
L = 42 - d - D. If you unfold the room this way,

                      H
                 o---------o
                 |         |
                 |      W/2|
                 |         |
                W|     F---o
                 |       D |
                 |         |
                 |         |
   o-------------o---------o-------------o---------o
   |      W      |    H    |      W      |    H    |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |L   Floor    |L        |L  Ceiling   |L        |L
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |      W      |    H    |      W      |    H    |
   o-------------o---------o-------------o---------o
   |             |
   |             |
   |H            |H
   |      S      |
   |     d|  W/2 |
   o------o------o
          W

and draw a diagonal line from S to F, this might be shorter than 
42 feet. The horizontal distance from S to F in this diagram is 
W/2 + (H-D), and the vertical distance is (H-d) + L + W/2. You figure 
out using the distance formula (or the Pythagorean Theorem) if this is 
possible.

If, however, you unfold the room this way,

                                  W
                           o-------------o
                           |             |
                           |     Fly     |
                           |H     F      |H
                           |      |D     |
                           |  W/2 |      |
   o-------------o---------o-------------o---------o
   |      W      |    H    |      W      |    H    |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |L   Floor    |L        |L  Ceiling   |L        |L
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |             |         |             |         |
   |      W      |    H    |      W      |    H    |
   o-------------o---------o-------------o---------o
   |             |
   |             |
   |H  Spider    |H
   |      S      |
   |     d|  W/2 |
   o------o------o
          W

and draw a diagonal from S to F, this also might be shorter than 
42 feet. The horizontal distance is now W/2 + H + W/2, and the 
vertical distance is (H-d) + L + D. Again, you should figure out using 
the distance formula (or the Pythagorean Theorem) whether or not this 
is possible.

HINT: Replace L by 42 - d - D first.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/