Distance between endpoints of parabola with length $80$

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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

$$l=2b$$