Equation of a catenary is
y = a * cosh(x/a) + C
I am told that two telephone poles are 10 m apart and that the poles are 6 m tall. The minimum clearance (distance of the cable to the ground) is 4 m. Question asks how long can the cable be.
So. I know that once I find the equation for the catenary the cable makes, I get the integral and use the arc-length formula. Unfortunately, I don't know how to get the 'a' in the formula.
Here's my attempt so far.
C doesn't really matter. The lowest point at x = 0 is a, since cosh(0) = 1. That means the point at x = -5 and x = 5 is a+2. So I get an equation:
a+2 = a * cosh(5 / a)
But I'm not sure how to solve it.
a * arccosh[(a+2) / a] = 5
That's as far as I got, and I'm rather sure I'm not doing it right. The assignment is focused on linearization and the like (Newton's method, fixed-point iteration, etc.).