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?

To find the point P that produces the crease of minimum length, we can use the concept of calculus and minimize the length of the crease using mathematics.

Let's consider the rectangular sheet of paper, where the width is 'a' and the length is 'b', and we want to fold it such that the crease length is minimized. We will place one corner of the sheet at point P on the opposite long side of the sheet.

To find the length of the crease, let's assume that point P is at a distance x from one of the corners on the long side. As the sheet is folded along the crease, we can imagine the crease as a line segment connecting the opposite corners of the sheet, and the crease length can be considered as the length of this line segment.

Now, let's consider the unfolded rectangle and draw a line segment connecting the opposite corners (which are the endpoints of the crease). This line segment divides the rectangle into two right triangles.

Using the Pythagorean theorem, we can find the length of the line segment/crease:
Length of the crease = √(a^2 + (b-x)^2).

To find the point P that minimizes the length of the crease, we need to differentiate the length equation with respect to x and set the derivative equal to zero.

Let's differentiate the length equation:
d/dx (√(a^2 + (b-x)^2)) = 0.

To simplify the differentiation, we can rewrite the equation as:
d/dx (a^2 + (b-x)^2)^(1/2) = 0.

By applying the chain rule of differentiation, we get:
(1/2) * (a^2 + (b-x)^2)^(-1/2) * (-2(b-x)) = 0.

Simplifying further:
-(b-x) / (a^2 + (b-x)^2)^(1/2) = 0.

Since a<b and 0<a, we can conclude that b-x ≠ 0. Therefore, the numerator -(b-x) must be zero:
b - x = 0,
x = b.

So, the point P that produces the crease of minimum length is at a distance of b from one of the corners on the long side.

Now, substituting x = b into the length equation, we get:
Length of the crease = √(a^2 + (b-b)^2) = √(a^2) = a.

Therefore, the length of the crease is 'a'.

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.