How far?

The distance from which a light can be seen is, in part, determined by its height. Knowing that the radius of the earth is about 6.4 103 km, how far could a light on a cliff 100 metres above sea level be seen?  Alternatively, how far is it from the light to the visible horizon?

[Hint: Because the cliff height is relatively small, take the distance to the actual light, rather than to the base of the cliff.]

My answer:

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Let the earth’s radius be R kilometers, and the height of the light h metres.

As in the adjacent diagram, we seek the length d of the third side of the right-angled triangle with hypotenuse of length R + h and side length R.

Then d 2 = (R + h)2 – R2 = 6400.12 – 64002

= 40 961 280.01 – 40 960 000

Taking the square root gives

                                d = 35.77 36 km.


This is an incorrect answer.

Did you use a right-angled triangle with one vertex at the centre of the earth?