Horizon

The top of a lighthouse is 19 m above the sea. How far away is an object which is just “on the horizon”? [Assume the earth is a sphere of radius 6378.1 km.]

Result

x =  15.6 km

Solution:

$R = 6378.1 \ km \ \\ h = 19 \ m = 0.019 \ km \ \\ \ \\ (R+h)^2 = R^2 + x^2 \ \\ x^2 = (R+h)^2 - R^2 \ \\ x^2 = R^2+2Rh + h^2 - R^2 \ \\ x^2 = 2Rh + h^2 \ \\ x = \sqrt{ 2Rh + h^2 } = 15.6 \ \text{km}$

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