Intermediate Value Theorem Problem on a String

[shinobi20]

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)

Preceding Problem. 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 $f(c) = c$.

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:

1. How can I describe the tangled string so that I can impose the preceding problem? The tangled string does not look like a function.
2. The preceding problem was shown in the general case, so is my statement correct at least for the case of $h(x)$ (right figure)?
3. 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.

 

[Office_Shredder]

I don't think I understand what h(x) actually represents, but I think you're trying to write down the wrong thing, or your plots are wrong In the previous problem you wanted to show two things were equal, so you wrote down a function which was the difference between the two of them. Here the thing you want is to show the distances from the left/right walls are equal.

 

[shinobi20]

I don't think I understand what h(x) actually represents, but I think you're trying to write down the wrong thing, or your plots are wrong In the previous problem you wanted to show two things were equal, so you wrote down a function which was the difference between the two of them. Here the thing you want is to show the distances from the left/right walls are equal.

The problem originally just showed the left figure, I added the right figure to show as another configuration of the tangled string. I represented the string on the right figure as $h(x)$, which is the parabolic-like string, it could be anything, I just drew some random curve instead of the ugly looking tangled string on the left figure to make my point. Also, the statement of the problem involves "at least one point of the wadded string whose distances from the two walls are exactly the same as they were before", it means that there should be at least a point in the tangled string (or the curve $h(x)$ I drew) such that its distance from the walls is the same as when that point is in the straight string before being thrown, it doesn't have to be equally distant from the walls.

 

[Office_Shredder]

The two things that need to be equal are:
Distance from the walls on the first string.
Distance from the walls on the second string.

Can you make some functions that describe these?

 

[shinobi20]

The two things that need to be equal are:
Distance from the walls on the first string.
Distance from the walls on the second string.

Can you make some functions that describe these?

I beg to disagree, the statement never said that the point must be equidistant from both walls, it just stated that its distance from both walls should be the same from when that point was still in the straight string, so that you can see in the right figure, both points align at $c$.

 

[hutchphd]

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)

whatever function describes the tangled string

The easiest choice for this function $h(x)=$ the horizontal displacement of the piece of the string originally at position x. Sketch the function for the tangle represented in your drawing. Is the answer clear now?

.

 

[shinobi20]

The easiest choice for this function $h(x)=$ the horizontal displacement of the piece of the string originally at position x. Sketch the function for the tangle represented in your drawing. Is the answer clear now?

.

I think that is just equal to the right figure I drew, the points are equal at $c$.

I wrote $h(x)$ just as an example but I need to show this for the general case where the string is tangled so I cannot really work with any function, I guess. Besides, the problem hinted to use the preceding problem (see original post) so I guess there must be some way to show that there exists a point by just arguing in accordance to the preceding problem.

 

[hutchphd]

Make a sketch of the function for the tangle you have drawn. Think about the end points for any such sketch

 

[Office_Shredder]

I beg to disagree, the statement never said that the point must be equidistant from both walls, it just stated that its distance from both walls should be the same from when that point was still in the straight string, so that you can see in the right figure, both points align at $c$.

I never said they had to be equidistant. Let me try again.

The distance from the left wall on the first string
The distance from the left wall on the second string.

Those two things need to be equal. Can you describe some functions that measure those?

 

[pasmith]

I would label a point on the string by $x \in [0,1]$. The distance to the left wall when straight is then $x$ and the distance to the left wall when tangled is $f : [0,1] \to [0,1]$. Thus, you want to show the existence of $x \in [0,1]$ such that $f(x) = x$, and you have $0 \leq f(0)$ and $f(1) \leq 1$.

 

[hutchphd]

I think that is just equal to the right figure I drew, the points are equal at c.

I think not. The displacement of point A on the string will be positive (to the right) . Th displacement of point B will by negative (to the left ) and the function is continuous. Somewhere in between the displacement must be...? QED

 

[shinobi20]

I would label a point on the string by $x \in [0,1]$. The distance to the left wall when straight is then $x$ and the distance to the left wall when tangled is $f : [0,1] \to [0,1]$. Thus, you want to show the existence of $x \in [0,1]$ such that $f(x) = x$, and you have $0 \leq f(0)$ and $f(1) \leq 1$.

So initially, we can model the straight string by labelling points on the string as $x$ and the distance of a point to the left wall be $f(x)$ for all $x$ (distance to the right wall is $1-f(x)$). To be specific, let $f(0)=0$ and $f(1)=1$, this also means that there is a point $c$ such that $f(c)=c$ since $f(0) \leq f(c) \leq f(1)$ and we have a $f(x) = x$ (this is not true if the string were not straight from the start) correspondence for the straight string.

Now, going to the tangled string. We can take $0 < f(0)$ and $f(1) < 1$ to represent the state of the tangled string. Since $f(x)$ is continuous, there exist a distance $f(a)$ such that $f(0) \leq f(a) \leq f(1)$ for all $a \in [0, 1]$. Using the conclusion of the "preceding problem", there exist a point $c \in [0, 1]$ such that $f(c)=c$. This shows that there exist a point $c$ which has the same distance $f(c)$ from the left wall as the a point in the straight line which is also labelled by $c$.

Is this correct?

 

[hutchphd]

And the displacement $h(x)=f(x)-x$ (which to me seems much easier to deal with) is as described in #11.

 

[Office_Shredder]

It doesn't have to be true that $f(0)\leq f(a) \leq f(1)$, and I don't think you actually need it to be true either.

 

[hutchphd]

But will be true that h(x)=0 for at least one x.