Chapter 4.1, Problem 62E

To determine

To find: the distance between the city SF with latitude $37°47^{\prime }36^{″}\text{ N}$ and city S with latitude $47°37^{\prime }18^{″}\text{ N}$ .

Answer to Problem 62E

Approximately 686 miles

Explanation of Solution

Given:

Radius of earth is 4000 miles. And both cities are on same longitude.

First step is to convert the latitudes from $D°M^{\prime }{S}^{″}$ to degree form.

  $\begin{array}{c}{\theta }_1=37°47^{\prime }36^{″}\text{ N}\\ \text{=}\left(\text{37+}\frac{47}{60}+\frac{36}{3600}\right)°\\ \approx 37.7933°\end{array}$

  $\begin{array}{c}{\theta }_2=47°37^{\prime }18^{″}\text{ N}\\ =\left(\text{47+}\frac{37}{60}+\frac{18}{3600}\right)°\\ \approx 47.6217°\end{array}$

Now, find the difference in latitudes of the two cities.

  $\begin{array}{c}\theta ={\theta }_2-{\theta }_1\\ =47.6217°-37.7933°\\ =9.8284°\end{array}$

Now, convert the degrees to radians.

  $\begin{array}{c}\theta =9.8284°\\ =9.8284°\times \frac{\pi }{180°}\\ \approx 0.1715\text{ radian}\end{array}$

Lastly, multiply the above radian with the radius of earth to find the arc length which corresponds to the distance between the two cities.

  $\begin{array}{c}s=r\theta \\ =4000\times 0.1715\\ \approx 686\end{array}$

Conclusion:

The distance between the two cities is approximately 686 miles.