Trying to prove the earth is flat

Geometry Level 1

In 1838, Samuel Birley Rowbotham conducted observations leading him to conclude the earth was flat. His idea was the following: If light goes straight and I'm standing at \(A\) with my friend, there must be a point \(B\) where I can't see him.

Assuming light does not bend because of atmospheric refraction, which of these is closest to the minimum arc length of \(\stackrel\frown{AB}\) that would prevent Samuel from seeing his friend? Assume they are both \(1.8\) meters tall and the radius of the earth is \(6,371,009\text{ m}.\)

Note: Though the earth is (of course) not flat, Samuel was able to see his friend at a farther distance than he was supposed to. This is due to atmospheric refraction, i.e. the non-homogeneous density of air creating a slight deviation of light. Without knowing this effect, a newspaper editor performed in 1896 the same experiment and concluded the earth was concavely curved.

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