Minimizing the Length of a Crease

Date: 11/11/1999 at 13:54:30
From: Jessica J. Johnston
Subject: Minimize the value of L

This is the exact problem in the book:
  
A rectangular sheet of 8.5 x 11-in. paper is placed on a flat surface, 
and one of the corners is lifted up and placed on the opposite longer 
edge. With all four corners now held fixed, the paper is smoothed 
flat.

a) Make the length of the crease as small as possible (call the length 
   of the crease L)

b) Show that L^2 = 2x^3/(2x-8.5)

c) Minimize L^2

d) Minimize the value of L

I'm stuck on (b) because I can't figure out how to show that it is 
true. I can't do the rest of them then either.

Date: 11/12/1999 at 17:06:00
From: Doctor Peterson
Subject: Re: Minimize the value of L

Hi, Jessica.

I like this problem, but because you didn't say what x was, I had to 
play with it for a day before I solved it. Every choice I made for x 
resulted in complex equations, until I chose the right one, for which 
your equation "fell out" of the drawing without too much work. But 
it's still not easy; you have to work through three different 
triangles and use Pythagoras twice to get the formula.


Here's my picture:

       D        E               A
        +-------+--------------+
        | W-x  / \      x      |
        |     /   \            |
        |    /     \           |
       y|   /x      \          |
        |  /         \         |
        | /           \        |
        |/            L\       |y+z
       F+               \      |
        |  \             \     |
        |     \           \    |
       z|         \        \   |
        |       y+z  \      \  |
        |               \    \ |
        |                   \ \|
       C+----------------------+B
        |           W          |
        |                      |

We want to find L in terms of x. F is the position of corner A after 
the fold, so EFB is a right triangle congruent to EAB. I've added a 
line BC parallel to AD, forming a rectangle, and called its length 
(the width of the paper, 8.5 inches) W. Then I labeled DE as W-x, and 
introduced variables y and z, whose sum is the length of AB, which 
you'll find on the way to the answer.

There are two important pairs of triangles. First, EAB and EFB are 
congruent, as I said, because they are the same part of the paper 
before and after folding. (You can make that more geometrical if you 
want.) That lets me label EF and BF as x and y+z. Second, EDF and FCB 
are similar; look at their angles.

You should first find y in terms of x using Pythagoras in triangle 
EDF. Then you can use the similar triangles to get z in terms of x. 
Finally, you can use Pythagoras to get L^2 in terms of x, y, and z, 
and substitute to get it in terms of x alone. There's some algebra 
involved, but everything simplifies neatly to give the formula you 
want:

     L^2 = 2x^3/(2x-W)

When you actually solve the problem, you'll want to check whether 
the solution is a crease that crosses the bottom of the paper, with 
y+z > 11. Unless that happens, the formula you get will be sufficient, 
but if it does, the crease will be shorter than the L you calculate.

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