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Question

one corner C of a very long strip of paper 10cm wide is folded so that, after being folded, the corner C lies on the long side of the strip opposite C. how should the paper be folded so as to minimize the length of the fold?

Answer 1

It is simplest to imagine setting up a coordinate system from the lower left corner and folding from the lower left to the right.

Let a be the horizontal distance to the crease (point A) and b be the vertical distance on the right side of the paper where the corner is placed (point B).

Then the distance from the origin O to A equals the distance from A to B. So

b^2 + (10-a)^2 = a^2. Multiply out and solve for a:

a=5 + (b^2)/20.

Now draw the perpendicular from B to the left side where it hits at point C = (0,c). The slope of line AB is b/(10-a). So the slope of line BC is -(10-a)/b. It goes through the point (10,b). So its equation is

y - b = -(10-a)(x-10)/b.

Now c is the y intercept of this line. So c = 50/b +b/2, when the formula for a in terms of b is substituted in and it is simplified. It is just c that we want as small as possible.

So we want to minimize this function of b. Its derivative is -50/b^2 + 1/2. Set it equal to 0 to get b = 10. This is how far up the corner should be on the other side, and when you think about it this is obviously the right answer.

Answer 2

You want to fold it so that C lines up with the opposite corner.

The fold cuts across the strip on a diagonal. The closer the fold is to being aligned perpendicular to the long sides of the strip, the shorter the fold will be in length. Which is the same thing as saying a line segment between points on two parallel lines is shortest when perpendicular to those lines.

Getting C to lie as far away as it can from its neighboring corner (10 cm away) is the way to do that.