mathschallenge.net

Problem

From how far on a clear day, and assuming there are no obstructions, could you see from the top of the Eiffel tower?

Take the height of the Eiffel tower as 300 metres and the radius of the Earth as 6240 km.

Solution

We shall answer this question in two ways:

Let us begin by considering the distance from the top of the tower to furthest point on the surface of the earth (the tangental length).

Applying the Pythagorean Theorem, 6240.3² = d ² + 6240². Hence d 61.2 km.

However, this does not represent the distance along the curvature of the earth (the arc length). To do this we apply trigonometry to the right angle triangle above, cos θ = 6240/6240.3, thus θ 0.562 degrees.

As the radius of the earth is 6240 km, the circumference is 2π 6240 39187 km. So the arc length will be 0.562/360 39187 61.2 km, which is the same answer as the other method... coincidence?

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