Hey guys I ran accross this problem while watching a YouTube video.
A cable of $80$ meters (m) is hanging from the top of two poles that are both $50$ m from the ground. What is the distance between the two poles, to one decimal place, if the center of the cable is:
(a) 20 m above the ground?
And yes, I did come across a solution involving hyperbolic trig but that's not what I am interested in.
I am interested in figuring out another way to solve this problem that does not involve using $sinh$. I am thinking that we can assume that this is a parabola because clearly there is a vertex there is symmetry. Is this assumption correct? Am I going to get anywhere with this assumption?
Best Answer
This is a classical problem. The curve is called a catenary, from the Latin word catena, meaning chain. The problem was already circulating around in the days of Galileo. Namely, people asked what shape will a chain take if we let it hang between two fixed points. Hence the name catenary. I read somewhere that Galileo himself thought it must be a parabola, but some other Italian mathematician proved it was not. So if you must make a mistake in a guess, better to make a mistake Galileo made too..
In short, there is no way to circumvent the hyperbolic cosine, because it is this function precisely that describes the catenary.
There is a youtube video that claims that the problem you mentioned was given in job-interview for Amazon. IMHO, this is a nice, albeit untrue, story.