# Length of an arc - when will a person leave your view?

Two people 1.8m tall walk from each other until they can no longer see each other (due to the curvature of the earth which has a radius of 6378km).

Assuming nothing else blocks their view, how far do they have to walk?

Note. I cant get my head around how this relates to what we've learned about radius and arc length.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

Let S be the distance between the two people such that after walking through a distance S, they won't be able to see each other.

Let R = 6378Km = 6378000 m be the radius of the earth.

Ler angle ACB = 2Theta (see figure)

We know that,

arc length / radius = angle in radians

Therefore, S/R = 2*theta

or, S = R * 2*theta ....(1)

Now, consider the triangle PCB. (The line CP divides S into two equal parts)

angle PCB = theta

PC = 6378000 - 1.8 = 6377998.2 m

CB = 6378000m

Cos[theta] = base/hypotenuse = 6377998.2/6378000 = 0.9999997

theta = arc cos[0.9999997] = 0.04 degrees

converting it to radians using

180 degree = pi radians

therefore, 0.04 degrees = pi*0.04/180 = 6.978*10^-4 radians

Therefore, theta = 6.978*10^-4 radians

Hence, arc length, S = R * 2 * theta = 6378000 * 2 * 6.978*10^-4

=> S = 8900.85 m

or S = 8.9 Km

That is after travelling a distance of 8.9 Km the two persons will not be able to see each other.

Hope you could follow well. The only concepts we used are the relation of arc length and radius of a circle and the basic trigonometric formulae.

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