Energy conserving path in the eerie conservation park

Calculus Level pending

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? Give the shortest longest distance that you have to walk (blindly) to reach that straight road.

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