Chapter 6.1, Problem 72E

To determine

Find the distance between two cities.

Answer to Problem 72E

  $s\approx 346mi$

Explanation of Solution

Given information:

Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of $35°N$ , and New Orleans has a latitude of $30°N$ . (The radius of the earth is $3960mi$ .)

Calculation:

Use trigonometry to calculate the distance between two points on the Earth’s surface if they lie on the same meridian and we are given the latitudes of the two points.

The length of an arc subtended by the angle $\theta$ is given by $s=r\theta$ , where $s$ is the length of the arc, $r$ is the radius of the circle, and $\theta$ is the angle in radians.

Also, remember that to convert degrees to radians we multiply by $\pi /180$ .

To convert radians to degrees, we multiply by the reciprocal, $180/\pi$ .

Now, given that the latitude of Memphis, Tennessee, is $35°N$ , and that of New Orleans, Louisiana, is $30°N$ ,

Given that the two cities lie on the same meridian and that the radius of the Earth is $3960mi$ ,

We can use the above relationships to calculate the distance between the two cities.

First we calculate the angle, $\theta$ . Between them in degrees. Hence

  $\begin{array}{l}\theta =35°-30°\\ \theta =5°\end{array}$

Next, we convert the angle to radians, so that

  $\theta =5°\times \frac{\pi }{180°}$

Put this into our equation, we get the arc that subtends angle $\theta$ , which is the distance between the two cities.

  $\begin{array}{l}s=\frac{5°}{180°}\pi \times 3960mi\\ s\approx 346mi\end{array}$