# Arc Length Problem

1. Feb 20, 2006

### SigurRos

Hey I got a project assigned for my Calc 3 class, and I was wondering what to do with the following:

A hanging cable has the shape

y = 1/c cosh(cx + b) + a

for some constants a,b,c with c>0. Suppose the ends are at P(0,10) and P2(30,5).

If the length of the cable is known to be 100 units then determine a,b,c and then plot the graph.

I know that dy/dx = sinh(cx + b), so the arc length formula would be:

100 = int(sqrt(1 + sinh(cx + b)^2)) from 0 to 30

but I'm having issues solving the equations in terms of a,b and c. I tried, and got an equation with lots of cosh's that myself and Maple could not solve or reduce.

Thanks alot!

2. Feb 20, 2006

### d_leet

you can simplify the
1 + sinh(cx + b)^2 into just (cosh(cx + b))^2.

And you were given 2 points which you can use to find equations relating a, b , and c.

Last edited: Feb 20, 2006
3. Feb 20, 2006

### SigurRos

Maple computed the derivative for me, and that is what it gave me.
Also, I did use those 2 points to construct 2 equations, but when I tried to solve for the system I got a complex equation with lots of cosh's that maple couldnt solve.

4. Feb 20, 2006

### d_leet

Ohh wow I feel dumb your deriavtive is right. I forgot to multiply by c..

5. Feb 21, 2006

### Pseudo Statistic

Newton's method?

6. Feb 21, 2006

### D H

Staff Emeritus
You are relying on Maple far too much. You should be able to compute the derivative by hand. You will need to guide Maple to find an approximate solution, with most of the work done by hand.

Hints:
1. Eliminate a.
2. Use the hyperbolic identities for cosh(u)-cosh(v) and sinh(u)-sinh(v).
3. Find tanh(15c+b).