# math (calculus)

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A rectangular sheet of paper of width a and length b, where 0<a<b, is folded by taking one corner of the sheet and placing it at point P on the opposite long side of the sheet. The fold is flatened to form a crease across the sheet. Assuming that the fold is made that there is no flap extending beyond the original sheet, find the point P that produces the crease of minimum length. What is the length of that crease?

• math (calculus) - ,

If we label the lower left corner A, and go clockwise for B,C,D

then with some actual folding experimentation, we see that P can go along side BC at a distance x from B, which can vary from b-sqrt(b^2-a^2) to a. Anywhere else the fold produces a flap that extends beyond the sheet.

When x = b-sqrt(b^2-a^2) the fold length is just the long side of the rectangle, b. That is, P is a distance sqrt(b^2-a^2) from C.

When x = a, the fold length is just the diagonal of a square, a√2

It's late. I'll let you figure out in which cases one is greater than the other.