The Hanging Cable: The Problem to Solve to Work at Amazon

Legend has it that this and similar problems have appeared in Amazon's internal interviews for software engineers and developers over the past few years. Whether that's true or not, we now have two poles that are 50 meters high, joined by a hanging cable that is 80 meters long. The center of the cable (and by extension, its closest point to the ground) is floating 20 meters above the ground. The question is: What is the distance between the two poles?

It's not the first time something like this happens. Sometimes job interviews simply seek to take candidates out of their element, evaluate their response speed, adaptability, creativity, and why not, their reaction to failure, a kind of mini-Kobayashi Maru if you will.

This time, what we have on hand brings a pinch of extra spice. A variant of the exercise is said to have appeared in an Amazon test for potential engineers, but it was thoroughly explored by Neil Chatterjee and Bogdan G. Nita in volume 4 of the Atlantic Electronic Journal of Mathematics in 2010:

The Problem

Two poles, joined by a hanging cable. Each pole is 50 meters high. The total length of the cable is 80 meters, and its lowest point is 20 meters above the ground. The question to solve is what distance separates the two poles. Many people try to follow exclusively the physics route, determining tension and curves, but at the same time mathematical solutions are needed, and I speak in plural because it requires more than one formula. But before that, split the graph in half, crossing vertically and horizontally at the lowest point of the cable.

The Solution

With this modified graph, things are a bit clearer: Our goal is to solve X and multiply by two. First we attack the highest point of the pole with:

Which with its assigned values and a relocation of the -a becomes:

And even deeper:

The second equation focuses on half the length of the cable:

Which we process a bit more:

Finally, we apply hyperbolic identity:

And we introduce the values:

Which gives us a value for a of 35/3, or 11.6 repeating. With the value of a established, we return to the second equation, inject the new parameters:

And we solve for x, which gives us 22.7 meters. If we remember, x represents half the distance, therefore the definitive answer is 45.4 meters. One thing is certain: I will never work for Amazon if they ask this. They can keep their poles and cables.