# Distance between endpoints of parabola with length $80$

I've seen a question asked in an interview as following.

How can be the distance indicated by question mark calculated? What are the ways?

The distance would be $$0$$ meters.

The ends of the parabola are $$50$$ meters high and the parabola itself $$80$$ meters in length. If you were to hold both ends of the parabola from the same point at that height, it would fall $$80/2 = 40$$ meters down, $$10$$ meters above the ground.

This is less any sort of involved detailed computation than it is a way to see how cleverly you can think.

• +1. "This is less any sort of involved detailed computation than it is a way to see how cleverly you can think." Indeed, the problem does not even indicate that the "curve" is actually a parabola; it could be a catenary (which would be appropriate if we're considering a rope or cable hanging across a chasm), or an arch of a sinusoid, or, or, or ... So, either the question is egregiously ill-posed (which is not unheard-of in interview questions or social media challenges) or it's being a little sneaky. Here, it's the latter. – Blue Jun 16 at 3:09
• only on 0 thickness rope. otherwise they lie about the length. – Roddy MacPhee Jun 16 at 14:10

Distance is $$0$$. The midpoint is at $$40$$ from the end. The midpoint height is $$10$$, so you need at least $$40$$ to get from top to middle, if the distance is $$0$$. If the distance is larger, then you need more cable.

hint

the equation of the parabola will be of the form $$y=ax^2+10$$ where $$-b\le x\le b,$$ $$ab^2+10=50$$

and

$$L=80=2\int_0^b\sqrt{1+4a^2x^2}dx$$ the length you look for is

$$l=2b$$