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.