Catenary - rope hanging between poles

[Jonnyb42]

Homework Statement

Problem is regarding approximating the value of a in y= a cosh(x/a) using Newton's method, and then use a to find the length of the rope.
That equation represents the curve formed by a rope hanging with it's ends attached to poles at a distance 2b.
(cosh() = hyperbolic cosine)

distance between poles = 2b = 400 ft.
sag = S = 30 ft.

Homework Equations

(previously proven eqns)

L = 2 a sinh(b/a)
where L is the length of the rope
a is a physical constant
and 2b is the distance between the poles that the rope is hanging from

S = a cosh(b/a) - a
where S is the sag (vertical distance between highest and lowest points on the rope)
a is a physical constant
2b is distance between poles

The Attempt at a Solution

I am really not sure how to do this, as I do not know how/why Newton's method could be used to get a value for a.
Also, they gave a hint, a clip from the Calc book I am reading: "[Hint: First let u = 200/a]"

 

[JaWiB]

How about f(a) = a cosh(b/a) - a - S

 

[Jonnyb42]

Is that correct? I didnt put that under at attempted solution because I thought it was not even right since I tried Newton's method on it and it wasn't really homing in on a particular number, or perhaps I messed up with something. I also thought I did it wrong because, first of all the book did not give a hint to how many iterations to do and my answer, after about 4 iterations, was 470 and the answer in the book was about 404 something.

 

[JaWiB]

Is that your answer for the value of a, or the value of L? Seems to me like it should work. Maybe you should post your attempt