Question

two cities are 1200 km apart and lie on the same north - south line. the latitude of the northernmost city is 26° n. what is the latitude of the other city? the radius of the earth is approximately 6400 km. (do not round until the final answer. then round to the nearest integer as needed.) the latitude of the other city is approximately □° n.

Explanation:

Step1: Recall arc - length formula

The arc - length formula on a circle is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. On the Earth, for two points on the same north - south line, the distance between them $s$ is related to the radius of the Earth $r$ and the difference in latitude $\Delta\theta$. Here, $s = 1200$ km and $r\approx6400$ km.

Step2: Solve for the central angle in radians

From $s = r\theta$, we can solve for $\theta$: $\theta=\frac{s}{r}$. Substituting $s = 1200$ km and $r = 6400$ km, we get $\theta=\frac{1200}{6400}=\frac{3}{16}$ radians.

Step3: Convert radians to degrees

We know the conversion formula from radians to degrees is $\theta_{degrees}=\theta_{radians}\times\frac{180^{\circ}}{\pi}$. So $\theta_{degrees}=\frac{3}{16}\times\frac{180^{\circ}}{\pi}$.
$\theta_{degrees}=\frac{3\times180}{16\pi}=\frac{540}{16\pi}\approx\frac{540}{16\times3.14}\approx10.74^{\circ}$.

Step4: Determine the latitude of the other city

If the northernmost city has a latitude of $26^{\circ}$ N, and the two cities are on the same north - south line, and the southern city is below the northern one. The latitude of the other city is $26^{\circ}-\theta_{degrees}$. So the latitude is $26 - 10.74\approx15^{\circ}$ N.

Answer:

$15$