Intermediate Value Theorem Word Problem

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SUMMARY

The discussion focuses on applying the Intermediate Value Theorem (IVT) to a problem involving a straight piece of string and a tangled string within a rectangular room. The user needs to demonstrate that there exists at least one point on the tangled string where the distances to two opposite walls are equal to those of the straight string. The approach involves defining the straight string as a function f(x) and the tangled string as g(x), leading to the formulation of h(x) = f(x) - g(x) to analyze the problem further.

PREREQUISITES
  • Understanding of the Intermediate Value Theorem (IVT)
  • Basic knowledge of function graphing and parameterization
  • Familiarity with distance calculations in a Cartesian coordinate system
  • Experience with mathematical problem-solving techniques
NEXT STEPS
  • Research the application of the Intermediate Value Theorem in geometric contexts
  • Explore parameterization techniques for curves and their implications
  • Learn about distance functions in coordinate geometry
  • Investigate methods for solving equations involving multiple functions
USEFUL FOR

Students studying calculus, particularly those tackling problems involving the Intermediate Value Theorem, as well as educators looking for examples to illustrate the theorem's application in real-world scenarios.

harrietstowe
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Homework Statement


The image can be seen at:
http://s1130.photobucket.com/albums/m521/harrietstowe/?action=view&current=photo1.jpg

The rectangle in the picture 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 blob is the same string tangled up. I need to show that there is at least one point of the tangled string whose distances from the two walls are exactly the same as they were before.

Homework Equations





The Attempt at a Solution


I tried to graph the situation by saying the regular string is f(x) and the tangled string is g(x) and then graphing h(x)=f(x)-g(x) but I am now stuck.
Thank You
 
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Think about the straight string of length L with x coordinate parameterized by arc length s so its x coordinate is f(s) = s, 0 ≤ s ≤ L. Then think of the tangled string still parameterized by s so

R(s) = < x(s), y(s) >, 0 ≤ s ≤ L

Then look at f(s) - x(s) and see what you come up with.
 

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