Chip Unicorn on Nostr: I have a question for mathematicians out there... Start with a square 100 km on a ...

I have a question for mathematicians out there...

Start with a square 100 km on a side.

An inhabitant of this square has two ways to travel: walking, which can be done from any point to any point at a pace $p_w$ seconds per kilometer, or going by a train which needs laid-down tracks but goes at a pace $p_t < p_w$ seconds per kilometer. The inhabitant can combine the two ways of travel -- for example, she could walk to the train, take the train, walk to another train, take that one, and walk to the final destination. If she walks 2 kilometers and takes the train 9 kilometers, her total time in travel would be $2p_w + 9p_t$.

Here's the catch: There's a limited amount of train track available. We can't just cover the square with train track.

If the start and end of the trip were uniformly randomly chosen after the train track layout were set, what layout of the track would minimize the expected time of the trip?

Would it be a grid with squares? Or something more interesting?

#mathstodon #geometry