Comment place poles 45.4m apart (Score 1) 283
Here's some R code to solve a). Place the poles 45.4m apart.
Graph of the solution: https://www.desmos.com/calcula...
It could be possible to solve this on paper with a huge mess of algebra. I took the easy way out with the R root solver, uniroot(). The trick is finding the k value. k ~=
# cable eqn is c(x) = (cosh(k*x) - 1)/k. passes through origin
# we need to know where c(x) = 30
f = function(x) { -40 + sinh(acosh(30*x + 1))/x }
# find the k that gives a length of 2*40 = 80m
k = uniroot(f,c(0.04,0.1),tol=1E-6)
L = (1/k$root) * sinh(acosh(30*k$root + 1))
a = acosh(30*k$root + 1)/k$root
cat("Place posts", 2 * a , "m apart\n")
cat("k =", k$root, "/m \n");