Question

What is the arc length formula and how would it be used to find the radius of the earth?

Answer 1

Hint: This problem deals with the arc length on any given radius of a circle. Here we have to obtain the arc length formula and we have to use this arc length formula to find the radius of the earth. We know that the radius of the earth is a big amount of measure, which is in fact a bigger unit of length measure. The radius of the earth is equal to 6378 km, we have to obtain this value approximately.

Complete step by step solution:
The arc length is the horizontal distance which is on the equator, between two locations with associated difference of longitude, assuming this longitude difference as \[\theta \], the value of \[\theta \] is equal to S km.
The formula used for the circular arc length is given below:
$ \Rightarrow $ Radius $ \times $ (angle subtended at the center in radian)
So the equatorial radius is given by as shown below:
$ \Rightarrow R = \left( {\dfrac{S}{\theta }} \right)\left( {\dfrac{{180}}{\pi }} \right)Km$.
Now let us consider two locations which are near the equator, which is Brazil.
The longitudes of Brazil are at ${45^ \circ }W$and ${55^ \circ }W$, so the longitudinal distance between these two longitudes is equal to $S = 1100Km$.
And the angle difference is $\theta = {55^ \circ } - {45^ \circ }$
\[\therefore \theta = {10^ \circ }\]
Now substituting these all values in the expression of the equatorial radius as shown below:
$ \Rightarrow R = \left( {\dfrac{{1100}}{{{{10}^ \circ }}}} \right)\left( {\dfrac{{{{180}^ \circ }}}{\pi }} \right)Km$
$\therefore R = 6302 Km$

Note: Please note that the actual value of the radius of the earth is equal to 6378 Km, but from the derived arc length formula, we have obtained the approximated value of radius of the earth. One more important thing to note is that we can take any arc length which is near to the equator, here we assumed Brazil.