Thought question - geometry/physics

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Let us consider a wooden plank that is placed on the earth's surface. Assuming the earth is a perfect sphere and the plank is infinitely long and massless, how would you determine the length of the plank at which you start feeling like you are on an incline, if we define that sensation occurring at 15 degrees? Your mass is negligible as well.

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Earth radius $ R_⊕ \approx 6371$ km

$R_⊕\tan 15° \approx 1707.1$ km