Question

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of ${\mathbf{35}}^{\mathbf{\circ }}\mathit{N}$, and New Orleans has a latitude of ${\mathbf{30}}^{\mathbf{\circ }}\mathit{N}$. Find the distance between these two cities. (The radius of the earth is 3960 mi.)

Short answer

The final distance between these two cities is approximate346 miles.

Step-by-step solution

Step 1. Given information.

Radius of the earth (r) $=3960mi$.

New or leans latitude$={30}^{\circ }N$

Memphis latitude$={35}^{\circ }N$

Step 2. Write the concept.

First, we find out central angle by using difference between latitudes.

Then convert value from degree to radian by multiplying $\frac{\pi }{180}$.

Now to find the distance between the two cities, we can use below formula:

$s=r\theta$

Where:

$s=$distance between cities.

$r=$radius of earth

$\theta =$central angle

Step 3. Determining the values.

The difference in latitudes gives the central angle of which is:

$$\begin{gathered} \theta ={35}^{\circ }-{30}^{\circ } \\ =5^{\circ } \end{gathered}$$

Now convert into radian by multiplying with $\frac{\pi }{180}$, we get:

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

$\theta =\frac{\pi }{36}$radians.

The length of an arc that subtends a central angle of $\theta$radians is:

$s=r\theta$

Put given values in the formula, we get:

$s=\frac{\pi }{36}\times 3960$

Put $\pi =3.14$in above expression, we get:

$$\begin{gathered} s=\frac{3.14}{36}\times 3960 \\ s\approx 345.575 \\ s\approx 346mi \end{gathered}$$