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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)

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Solution Solution | Comment 1 of 2

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

  Posted by Bractals on 2008-10-16 15:36:53
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