You are stranded in the middle of a large desert and the only way home is a through a straight road, which unfortunately you do not know the location of.
If the perpendicular distance from you to the road is $x$ kilometres (you do not know what $x$ is), what is the smallest possible distance (in terms of $x$) you need to travel to ensure you will reach the road?
(I was thinking of maybe travelling in a spiral but I really don't have any idea.)
This general type of problem is called a "lost in the forest problem," and precise solutions depend crucially on the geometry of the forest (or, in your case, the desert.) You can find more information by Googling that term.
The usual versions of the problem deal with a bounded forest that you're trying to reach the edge of. If I understand your problem correctly, you have an unbounded desert and the boundary is a thin strip in the middle. But it's ultimately the same type of problem.