Minimizing the Length of a CreaseDate: 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/ |
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