Chapter 6.1, Problem 102AYU

To determine

To find: the distance between two cities Charleston and Jacksonville.

Answer to Problem 102AYU

The distance between two citiesis $s\approx 554\text{miles}$

Explanation of Solution

Given information:

The given radius of earth is $\text{3960}\text{miles}$

The angle formed by a ray drawn from the centre of Earth to the equator is ${\text{38}}^{\text{o}}\text{21'}\text{north}\text{latitude}$

The angle formed by a ray drawn from the centre of Earth to latitude of a location ‘L’ is ${30}^{\text{o}}\text{20'}\text{north}\text{latitude}$

Concept used:

The object travels around the circles;

That an object moves around a circles of radius $r$ at a constant speed .If $s$ is the distance travelled in time $t$ around this circles, then the linear speed $v$ of the object its defined as.

  $v=\frac{s}{t}$

The relation between linear speed and angular speed;

  $v=r\omega$

Here $\omega$ is measured in radius per unit time.

The angular speed $\omega$ of this object is the angle $\theta$ (measures in radians) swept out. Divided by the elapsed time $t$ that is

  $\omega =\frac{\theta }{t}$

The radius of earth is

  $r=3960\text{miles}$

Convert the angle of ${\theta }_1={38}^{\circ }21\text{'}$ minuteto degree.

  $\begin{array}{c}{38}^{\circ }{\text{21'=38}}^{\circ }+21\times {\left(\frac{1}{60}\right)}^{{}^{\circ }}\\ \text{=38}{.35}^{\circ }\end{array}$

Convert the angle of ${\theta }_2={30}^{\circ }20\text{'}$ minute todegree

  $\begin{array}{c}{30}^{\circ }{\text{20'=30}}^{\circ }+20\times {\left(\frac{1}{60}\right)}^{{}^{\circ }}\\ \text{=30}{.33}^{\circ }\end{array}$

The measure of the central angle between the two cities is,

  $\begin{array}{c}\theta ={\theta }_1-{\theta }_2\\ \theta \text{=38}{\text{.35}}^{\text{o}}\text{-30}{\text{.33}}^{\text{o}}\\ \text{=8}{\text{.02}}^{\text{o}}\end{array}$

Convert the angle of $\theta ={8.02}^{\circ }$ degree to radian

  $\begin{array}{c}\theta ={8.02}^{\circ }\\ =8.02\times 1\text{degree}\\ \left(1\text{degree}=\frac{\pi }{180}\text{radian}\right)\\ \theta =8.02\times \frac{\pi }{180}\text{radian}\\ \theta =0.1399\text{radian}\end{array}$

The distance between the two cities is.

  $s=r\theta$

The radius is $3960\text{miles}$ and the central angle is $\theta =0.1399\text{radian}$

The distance between the two citiesis calculated as,

  $\begin{array}{c}s=3960\times 0.1399\\ \text{=554}\text{.004}\text{miles}\end{array}$

Therefore, the distance between two cities Memphis and Louisiana is $s\approx 554\text{miles}$