All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
maximum square folding (Posted on 2008-10-16) Difficulty: 2 of 5
Start with a square piece of paper. Label the vertices ABCD. Pick a point on CD and label it E. Fold along the line BE. Label the new location of C as C'. Find the point F on AD such that when folding along BF it makes the new location of A coincide with C'. Now lastly find a point G on AD such that when folding along EG it makes the new location of D lie on EF (either EC' or A'F). After all 3 of these folds are completed you should have a new irregularly shaped quadrilateral FBEG.

For simplicity's sake assume the original square is of unit length. Now the 2 problems are:

1) If x is the length of CE, then give an equation for the area of FBEG based on x.

2) Find the x that maximizes the area of FBEG

See The Solution Submitted by Daniel    
Rating: 2.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Solution | Comment 1 of 2

Let [PQ...W] denote the area of polygon PQ...W.
Then
   [FBEG] = [ABCD] - [ABF] - [BCE] - [EDG]
Finding the segment lengths in terms of x:
   |ED| = 1-x
   |DG| = x(1-x)
           1-x
   |FA| = -----
           1+x
Therefore,
                  1-x       x     x(1-x)^2
   [FBEG] = 1 - -------- - --- - ----------
                 2(1+x)     2         2
             1+x+x^3-x^4
          = -------------
               2(1+x)
Finding the x that maximizes [FBEG]:
   [2(1+x)][1+x+x^3-x^4]' = [1+x+x^3-x^4][2(1+x)]'
                         or
    [2(1+x)][1+3x^2-4x^3] = [1+x+x^3-x^4][2]
                         or
           x^2(3x^2+2x-3) = 0
Thus,
        sqrt(10)-1
   x = ------------
            3
 

  Posted by Bractals on 2008-10-16 15:36:53
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information