Q. 102 - Precalculus Enhanced with Graphing Utilities

Question

Charleston, West Virginia, is due north of Jacksonville, Florida. Find the distance between Charleston (${38}^{\circ }21\text{'}$ north latitude) and Jacksonville (${30}^{\circ }20\text{'}$ north latitude). Assume that the radius of Earth is $3960$ miles. 

Step-by-Step Solution

Answer

The distance between two cities is $554.5$ miles.

Step 1. Given information

Latitudes of two cities are given as ${30}^{\circ }20\text{'}$ and ${38}^{\circ }{21}^{\text{'}}$

The radius of the Earth is given as $3960$ miles.

To solve the problem, we assume the Earth as a circle. 

Step 2. Required formulae

• $1\text{'}={\frac{1}{60}}^{°}$
  
• A circle with radius $r$has the length of the perimeter $2\mathrm{\pi r}$
• The arc length of a circle made by an angle $\theta$and the radius $r$ is $S=r\theta$

Step 3. Calculation

The angle made by the difference of the two cities is, 

$$\begin{gathered} \theta =38°21\text{'}-30°20\text{'} \\ =(38+\frac{21}{60})°-(30+\frac{20}{60})° \\ =38.35°-30.33°\approx 8.02°=0.14radians \end{gathered}$$

Therefore, the distance between the two cities is the arc length made by this angle and the radius of the Earth, which is $s=0.14\times 3960\approx 554.5$ miles