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 6400 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. Given the first - city's latitude $\varphi_1 = 22^{\circ}\text{N}$ and the second - city's latitude $\varphi_2=31^{\circ}\text{N}$, then $\theta=\vert\varphi_2 - \varphi_1\vert=\vert31 - 22\vert = 9^{\circ}$.

Step2: Convert the angle to radians

Use the conversion formula $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$. Substituting $\theta_{deg}=9^{\circ}$, we get $\theta_{rad}=\frac{\pi}{180}\times9=\frac{\pi}{20}$ radians.

Step3: 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 $r = 6400$ km and $\theta=\frac{\pi}{20}$ radians. Then $s=6400\times\frac{\pi}{20}$.
Calculate $s = 320\pi\approx320\times3.14159 = 1005.31$ km.

Answer:

$1005$