energy-conserving path in the eerie conservation park?

M

Mugen is Strong

New member
After a Rip-van-Winkle-like sleep, you wake up in the forest. At your feet is a signpost that indicates that it is one mile to the road, and you know that this road is the only one in the area and that it runs straight. Unfortunately, the signpost has fallen down, so you know only the distance to the road, not the direction. What search path should you follow so that the worst-case distance you go before finding the road is as small as possible?
 
That's funny, I saw this problem before recently, and someone posted an answer of 2 + (3/2)?, which is the distance of a path wherein one walks straight out 1 mile, thence walks 3/4 of a circle, and then walks one more mile parallel to the first mile. That seemed like the right answer, but there is a shorter way. One should walk 2/?3 of a mile, thence another 1/?3 mile back at a angle of 60° to the first segment, then around 7/12 of a circle, and then 1 more mile tangent to the end of the circle, for a total distance of 1 + ?3 + (7/6)? which is just a bit shorter, about 6.39724, instead of 6.71239.

Can anyone beat this?

Edit: One way to find this angle of 60° is to work out the length as a function of the angle the first segment makes with the road that the last segment is perpendicular to, and find the minimum length. A briefer way might be to observe that 1) any straight line segment is always a tangent of the circle segment that it meets with, and 2) the 1st and 2nd segments makes the same angle with that road at equal angles, like a reflected light ray. Armed with 1) and 2), it's not hard to show that the angle has to be 60°. Maybe I should post a graphic?
 
I don´t know but I know what I would do.
I would put sticks in the ground a bit away from the sign, all at the same distance from the sign. (I would measure it someway, that can be done.)
After that I would continue doing so from the circle I had formed, to be sure that next
circle would be the exact distance from a starting point at the circle I just made.
(Sigh! Did you get that?......)
In the end I will have gone one mile in all directions around the fallen sign.
 
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