Calc 3 Project: Solve Arc Length Problem with y = 1/c cosh(cx + b) + a

[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.

Any advice?

Thanks alot!

 

[d_leet]

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.

Any advice?

Thanks alot!

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.

 

[SigurRos]

You forgot a 1/c in your derivative

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 couldn't solve.

 

[d_leet]

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 couldn't solve.

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

 

[Pseudo Statistic]

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 couldn't solve.

Newton's method?

 

[D H]

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 couldn't solve.

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).