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.
(In reply to Not there yet.
Following armando's hint, but keeping to my idea of walking a bit past the circumference of the one kilometer radius circle, finding a tangent line to that circle and walking it and proceeding around the circle to some point where a same length line, tangent to the circle, will reach the same distance from the circle as the original distance past such that the line between those two points are also tangent to the circle.
Using the equations that are used for polygons in calculating side, apothem, radius, etc., yet iterating through fractional number of sides, I find that the radius (the initial distance of the first leg of the trek is approximately 1.242003574189 km with second and last legs of the trek being approximately 1.473190928968 km, with the arc length of the circle trekked, for a total approximate distance of 6.458912350348 km. This is close (though slightly higher) to armando's calculation, but that may be due to precision of calculations (I calculated that the pentagon-version of the trek would be ~ 6.459064217818 km. The smaller figure I found is using the formula as if the polygons had ~ 4.948 sides.]
Edited on January 9, 2016, 11:50 pm
Posted by Dej Mar
on 2016-01-09 23:39:14