Can we minimize $\sqrt{x^2-120x+7200}+\sqrt{x^2+1600}$ when $0\le x\le 60$ without using calculus? It is easy to show using calculus that this function has a minimum at $x=24$ and the minimum value is $20\sqrt{34}$, but I have no idea how to approach this is an elementary way.
I would also like to refrain from using analytic geometry, since I came across this when trying to solve a middle school problem for a cousin. I am not really sure whether we can minimize that function with these "restrictions", I didn't specifically encounter it in an exercise, I just reduced an exercise to this.
