How Far Away is the Horizon?

It was the first time that Poole had seen a genuine horizon since he had come to Star City, and it was not quite as far away as he had expected.... He used to be good at mental arithmetic--a rare achievement even in his time, and probably much rarer now. The formula to give the horizon distance was a simple one: the square root of twice your height times the radius--the sort of thing you never forgot, even if you wanted to...

--Arthur C. Clarke, 3001, Ballantine Books, New York, 1997, page 71

In the above passage, Frank Poole uses a formula to determine the distance to the horizon given his height above the ground.

1. Use algebraic notation to express the formula Poole is using.
2. Beginning the diagram below, use one of the circle theorems to derive your own formula. You will need to add some more elements to the diagram.

   horizon1.eps

3. Compare your formula to Poole's; you will find that they do not match. How are they different?
4. When I was a boy it was possible to see the Atlantic Ocean from the peak of Mt. Washington in New Hampshire. This mountain is 6288 feet high. How far away is the horizon? Express your answer in miles. Assume that the radius of the Earth is 4000 miles. Use both your formula and Poole's formula and comment on the results. Why does Poole's formula work so well, even though it is not correct?