Mark Asks for Help
On a Internet message board, "Mark" posted the following:
Can you help Mark?
NOTE: Make sure you look at the hint.....
"I hoping one of you smart people can help me. here's the problem.
You have a sheet of paper 25 units long by 6 wide. With the long side facing you, fold the upper right corner to touch the bottom, so that the crease is minimized. Write a function defining the crease, and find the minimum."
Note: This is my visual interpretation of Mark's problem:
Hint: Mark even gave a hint: "This will take geometry or trig as well as calculus and creativity."
Sorry, that's all he said!
But I will add that one person (TM) has already responded to this week's posing of the problem with some concerns and great insights: "I've been playing with the crease problem on your website and I'm not sure if I understand the problem correctly. The problem, as I understand it, is to find the shortest crease. Wouldn't that be when the paper is folded in half? When the top right corner touches the bottom right corner the crease length would then be six in this case.
What I have been trying to do is find a function that would define the crease as the top right corner moves from the bottom left to the bottom right. Then finding the minimum of that function should result in a nice six. So far many pieces of paper have fallen victim to this attempt of mine. One interesting thing that I noticed was no matter where the top right corner lays on the bottom of the paper there are always two or three similar triangles. For example if you take the bottom of the crease, resulting from the top right to bottom left fold, and the two bottom points as the vertices of a triangle and rotate that 90 degree counter clockwise there is another triangle similar to it. flip the paper to see it better. As you move the top right point along the bottom edge from the left to the right another triangle appears to the left of the top right point, this triangle is also similar to the other two triangles. One other point of interest is that no matter where the top right corner lays on the bottom edge if you extend the "line", or segment, that is the left edge of the paper and extend the segment that results in the top right corner touching the bottom edge this will form an isosceles triangle that goes to infinity as the top right corner goes to the bottom right corner.
Solution Commentary: And Mark's reflection on a solution:
"Here's what I did, but I'd like another opinion(s).
The crease is shortest when the upper right corner is folded all the way to the lower left corner. A right triangle can be drawn with a height of 6, the base is unknown, and the crease is the hypotenuse. I think the triangle is similar to the triangle if you draw a diagonal across the paper before folding. Therefore 6/25 = side/6 the side is 36/25. Then I used the pythagorean theorem and said crease -sqrt(6^2+(36/25)^2)
The crease comes out to about 6.17 units. I made a scale model, and that's pretty close. Do you think it's right?"
Do you agree....is Mark trying to find the min or the max crease length? And what triangles are similar? A lot to play with here before we rush to tell Mark....Also, he originally posted his request on November 6, 1999, and no one else has responded yet!
Also, it seems Mark may have read the original problem wrong or is unaware of the rich tradition for this problem.