Lost in Illinois

Geometry Level 5

You are lost in a cornfield in Illinois. You do not know what shape the cornfield is, but you know that it is convex and has an area of 120π \frac{120} { \pi} meters2^2. Because the corn is so tall and dense, you are unable to see anything around you, so you are unable to see where the boundary of the cornfield is until you have crossed it. While standing in the cornfield, you think of a path S that, when you walk along it, will be guaranteed to get you out of the cornfield at some point, no matter what shape the cornfield has. Let ss be the minimum possible length the path S could have. What is the value of s2s^2?

Details and assumptions

You may use the fact that the maximal area that can be enclosed in a curve of length 2πR 2 \pi R is πR2\pi R^2, i.e. the circle.

If your strategy is simply to walk east, then you could be stuck in the west end of a rectangular cornfield with dimensions 120×1π 120 \times \frac{1}{\pi} , and so you could need to walk nearly 120 meters.

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