You are driving along a perfectly straight road through the woods and decide the trees look like an inviting hike. After all, you have a GPS that could easily get you back to the road. So you get out and head off in a straight line perpendicular to the road, not paying any attention to your direction because, hey, GPS.
Unfortunately after traveling 1km your GPS crashes. It loses all of its map data as well as any previous journeys. In you panic, you even forgot which direction you were walking.
So here you are: 1 km away from the long straight road (the only one around for many km) in an unknown direction. You have a GPS that can still give your accurate position and path relative to your start.
What is the length of the shortest path (measured from here) that guarantees you will find the road?
Note: the trees are dense enough that you could be very close to the road and not see it.
With the hint of Jer I realize that I was missing something. So i see now that you don't need symmetry in the path. Once you have cover the 3/4 of the circle you can leave the circle and go straight for 1 km. This really cut some 40 metres.
On the other side i've try to refine a little more. The equation to minimise the path isn't so easy but has an easy solution (the angle between the first direction taken and the radio where the walker begin his walk on the circle is 30°).
So you should begin walking for 1.154 km then turn on the left an angle of 60° (i mean turning 300°) and walk for 0.5772 km. Now you are on the circle, go ahead for 3,665 km on the circle and maintain your last direction for one more km. At that point you have been walking for 6.396 km, and you would have at some point found the road.
(I'm editing it again, cause of an error).
Edited on January 10, 2016, 6:54 am
Edited on January 10, 2016, 10:09 am
Posted by armando
on 2016-01-10 06:27:17