- #1

shinobi20

- 271

- 20

- Homework Statement
- The rectangle in the figure (left) represents the floor of a room, and AB a straight piece of string lying on the floor whose ends touch the opposite walls W1 and W2. The tangle represents the same piece of string wadded up and thrown back down on the floor. Show that there is at least one point of the wadded string whose distances from the two walls are exactly the same as they were before. Hint: See the preceding problem.

- Relevant Equations
- Intermediate value theorem (IVT)

**Let ##y=f(x)## be a continuous function defined on a closed interval ##[0, b]## with the property that ##0 < f(x) < b## for all ##x## in ##[0, b]##. Show that there exist a point ##c## in ##[0, b]## with the property that ##f(c) = c##.
Preceding Problem.**

This problem can be solved by letting ##g(x) = f(x) - x## so that ##-x < g(x) < b-x##. Since ##x \in [0, b]## is positive, ##g(x)## varies from some negative values to some positive values. Using the intermediate value theorem, we can conclude that ##g(c) = 0## which implies ##f(c) = c##.

Going back to the original problem which hinted to use this preceding problem, my initial idea is to use the straight string AB as the interval ##[0, b]## so that whatever function describes the tangled string (I don't think it looks like a function but whatever describes the points of the string) it could be mapped to a point in the straight string AB.

It is intuitive that another possible configuration of the string after it lands on the floor could be seen from the figure on the right. Let ##h(x)## be the function that describes the parabolic-like curve of the string. You can see that the point ##c## matches a point on the parabolic-like curve of the string which is the point having the same distance from the walls W1 and W2 as the point ##c##.

Since ##h(x)## is a function that is bounded above by the height of the wall (where the string AB is hanged), then there should exist a ##c## such that ##h(c) = c##.

Questions:

- How can I describe the tangled string so that I can impose the preceding problem? The tangled string does not look like a function.
- The preceding problem was shown in the general case, so is my statement correct at least for the case of ##h(x)## (right figure)?
- Can anybody guide me on this problem, giving hints on how to approach this? I don't want complete solutions since that will be a waste of this problem.