Question

two cities have nearly the same north - south line of 99° w. the latitude of the first city is 27° n, and the latitude of the second city is 34° 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: Calculate the difference in latitudes

The first - city latitude is $27^{\circ}$N and the second - city latitude is $34^{\circ}$N. The difference in latitudes $\Delta\theta=34 - 27=7^{\circ}$.

Step2: Convert degrees to radians

We know that to convert degrees to radians, we use the formula $\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}$. So, $\Delta\theta_{rad}=\frac{\pi}{180}\times7$.

Step3: Use the arc - length formula

The arc - length formula for a circle is $s = r\theta$, where $s$ is the arc - length (distance between the two cities), $r$ is the radius of the circle (radius of the Earth, $r = 6400$ km), and $\theta$ is the central angle in radians. Substituting $r = 6400$ km and $\theta=\frac{\pi}{180}\times7$ into the formula, we get $s=6400\times\frac{\pi}{180}\times7$.
\[

$$\begin{align*} s&=6400\times\frac{7\pi}{180}\\ &=\frac{6400\times7\pi}{180}\\ &=\frac{44800\pi}{180}\\ &\approx\frac{44800\times3.14}{180}\\ &=\frac{140672}{180}\\ &\approx782.62 \end{align*}$$

\]

Answer:

783