Question

two cities have nearly the same north - south line of 102° w. the latitude of the first city is 22° n and the latitude of the second city is 31° n. approximate the distance between the cities if the average radius of earth is 6402 km. the cities are approximately kilometers apart. (round to the nearest integer as needed.)

Explanation:

Step1: Find the central angle

The central angle $\theta$ between the two cities is the difference in their latitudes.
$\theta=(31 - 22)^{\circ}=9^{\circ}$
Convert degrees to radians: $\theta = 9\times\frac{\pi}{180}=\frac{\pi}{20}$ radians.

Step2: Use the arc - length formula

The arc - length formula is $s = r\theta$, where $r$ is the radius of the Earth and $\theta$ is the central angle in radians.
Given $r = 6402$ km and $\theta=\frac{\pi}{20}$
$s=6402\times\frac{\pi}{20}$
$s=\frac{6402\pi}{20}\approx\frac{6402\times3.1416}{20}$
$6402\times3.1416 = 20110.2432$
$\frac{20110.2432}{20}=1005.51216$

Answer:

$1006$