Hyperbolic functions are usually studied in a firstyear calculus course. Just about
every calculus book has an example or problem that uses the hyperbolic cosine function to model the shape of
a hanging cable (power line, chain). Physics gives us the following model:
where T is the tension in the cable at its lowest point and w is the weight of the
cable per unit length.
The shape of the curve of the hanging cable is called a catenary.
How accurate is this model? We performed the following experiment to find out.
We had two of our students hold a chain in the front of the classroom. Using a digital camera, we took a picture of
the hanging chain. (You can also hang the chain between two fixed supports). We downloaded the picture into our
computer and used a picture viewing software package to display the picture. Using copy and paste, we copied the
picture and pasted it on top of the axes plotted in TEMATH's Graph window.
Using TEMATH's Selection tool on
the tool palette and the arrow keys on the keyboard, we repositioned the picture so that the lowest point on the hanging
chain was placed at the origin. To see the entire chain picture,
Since the yintercept of the model curve y = (T/w) cosh(wx/T) is equal to T/w, we
need to shift the model down T/w units so that it passes through the origin (that is where we placed the
"yintercept" of the picture of the hanging chain). Thus, our model becomes
y = (T/w)(cosh(wx/T)  1).

Finally, we replaced the quantity (T/w) with the single parameter A to get
To determine how well this theoretical function models the hanging chain, we need to
find the value of the parameter A. We used the "Estimate and Improve" method.
The Work window contains the initial estimate A = 1.25.
 Select y1(x) in the Work window.
 Select Plot from the Graph menu.
Note that the curve does not match the shape of the hanging chain very well. We
need to improve our estimate of A.
 Change the value of A in the Work window.
 Select y1(x) in the Work window.
 Select Plot from the Graph menu.
Continue to change (improve) the value of A until you find a perfect match between the
curve of the model and the shape of the hanging chain.
Copyright 20002008 Adam O. Hausknecht and Robert E. Kowalczyk