We split the 80 m cable in half and work 1/2 of the problem
y = a Cosh (x / a) -a
We draw x and y axis
coordinate of top of the pole would be (x, 30)
30 = a Cosh(x / a)-a
first equation: (30 – a) / a = Cosh(x/a)
a sinh (x/a) = 40
second equation = sinh(x/a) = 40 / a
We use : cosh 2 (t) – sinh 2 (t) = 1
(( 30 + a) / a ) 2 – (40 / a) 2 = 1
(30 + a) 2 – (40 ) 2 = a 2
a = 11.67
11.67 * sinh( x / 11.67) = 40
sinh( x / 11.67) = 3.427
asinh( sinh ( x / 11.67) = asinh( 3.427 )
x / 11.67 = 1.945
x = 22.7 m is one half of the value. The poles are 45.4 meters apart
Amazon did a similar problem to hire you. The cable is 10 m from the center point to ground. When you split the cable in half, you have 40 m. So the bottom of the cable is 10 m from the ground. The solution is 0 m apart because the cable would be on the same pole.