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.]

Correct result:

x =  15.6 km

Solution:

R=6378.1 km h=19 m=0.019 km  (R+h)2=R2+x2 x2=(R+h)2−R2 x2=R2+2Rh+h2−R2 x2=2Rh+h2 x=2Rh+h2=15.6 kmR = 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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