Chapter 4.1, Problem 61E

To determine

To find: the distance between the city D with latitude $32°47^{\prime }{9}^{″}\text{ N}$ and city O with latitude $41°15^{\prime }50^{″}\text{ N}$ .

Answer to Problem 61E

Approximately 592 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.

  $\begin{array}{c}{\theta }_1=32°47^{\prime }{9}^{″}\\ \text{=}\left(\text{32+}\frac{47}{60}+\frac{11}{3600}\right)°\\ \approx 32.7864°\end{array}$

  $\begin{array}{c}{\theta }_2=41°15^{\prime }50^{″}\text{ N}\\ =\left(\text{41+}\frac{15}{60}+\frac{50}{3600}\right)°\\ \approx 41.2639°\end{array}$

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

  $\begin{array}{c}\theta ={\theta }_2-{\theta }_1\\ =41.2639°-32.7864°\\ =8.4775°\end{array}$

Now, convert the degrees to radians.

  $\begin{array}{c}\theta =8.4775°\\ =8.4775°\times \frac{\pi }{180°}\\ \approx 0.14796\text{ radian}\end{array}$

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

  $\begin{array}{c}s=r\theta \\ =4000\times 0.14796\\ \approx 591.84\end{array}$

Conclusion:

The distance between the two cities is approximately 592 miles.