Question

Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lieapproximately on the same meridian. Pittsburgh has a latitudeof $\mathbf{40}{\mathbf{.5}}^{\mathbf{\circ }}\mathit{N}$, and Miami has a latitude of 25.5 N. Find the distance between these two cities. (The radius of the earth is3960 mi.)

Short answer

The final distance between these two cities is approximate 1037 miles.

Step-by-step solution

Step 1. Given Information.

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

Pittsburgh latitude $={40.5}^{\circ }N$

Miami latitude$=25{.5}^{\circ }N$

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:

$$\begin{gathered} \theta ={40.5}^{\circ }-{25.5}^{\circ } \\ ={15}^{\circ } \end{gathered}$$

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

$$\begin{gathered} \theta ={15}^{\circ }\times \frac{\pi }{{180}^{\circ }} \\ =\frac{\pi }{12}radians \end{gathered}$$

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

$s=r\theta$

Put given values in the formula, we get:

$$\begin{gathered} s=\frac{\pi }{12}\times 3960 \\ \approx 1036.7255 \\ \approx 1037\text{miles} \end{gathered}$$