Q. 101 - Precalculus Enhanced with Graphing Utilities

Question

Memphis, Tennessee, is due north of New Orleans, Louisiana. Find the distance between Memphis (${35}^{°}{9}^{\text{'}}$ north latitude) and New Orleans (${29}^{°}{57}^{\text{'}}$north latitude). Assume that the radius of Earth is $3960$ miles.

Step-by-Step Solution

Answer

The distance between the two cities is $359.5$ miles.

Step 1. Given information

Latitudes of two cities are given as ${29}^{\circ }57\text{'}$ and $35°9^{\text{'}}$

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,

 $$\begin{gathered} \theta =35°9\text{'}-29°95\text{'} \\ =(35+\frac{9}{60})°-(29+\frac{95}{60})° \\ =35.15°-29.95° \\ =5.2° \\ =5.2\times \frac{\mathrm{\pi }}{180}\approx 0.09radians \end{gathered}$$

Therefore, the distance between the two cities is the arc length made by this angle and the radius of the Earth, which is $S=3960\times 0.09\approx 359.5$ miles