Chapter 6.1, Problem 101AYU

To determine

To find:find the distance between two cities Memphis and Louisiana.

Answer to Problem 101AYU

The distance between two citiesis $s\approx 359\text{miles}$

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.

  ${\text{35}}^{\text{o}}\text{9'}\text{north}\text{latitude}$

The angle formed by a ray drawn from the centre of Earth to latitude of a location ‘L’ is

  ${\text{29}}^{\text{o}}\text{57'}\text{north}\text{latitude}$

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 intime $t$ 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 theangle $\theta$ (measures in radians)sweptout. Divided by the elapsed time $t$ thatis

  $\omega =\frac{\theta }{t}$

The radius of earth is

  $r=3960\text{miles}$

Convert the angle of ${\theta }_1={35}^{\circ }9\text{'}$ minute to degree.

  $\begin{array}{c}{35}^{\circ }{\text{9'=35}}^{\circ }+9\times {\left(\frac{1}{60}\right)}^{{}^{\circ }}\\ \text{=35}{.15}^{\circ }\end{array}$

Convert the angle of ${\theta }_2={29}^{\circ }57\text{'}$ minute todegree

  $\begin{array}{c}{29}^{\circ }{\text{57'=29}}^{\circ }+57\times {\left(\frac{1}{60}\right)}^{{}^{\circ }}\\ \text{=29}{.95}^{\circ }\end{array}$

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

  $\begin{array}{c}\theta ={\theta }_1-{\theta }_2\\ \theta \text{=35}{\text{.15}}^{\text{o}}\text{-29}{\text{.95}}^{\text{o}}\\ \text{=5}{\text{.2}}^{\text{o}}\end{array}$

Convert the angle of $\theta ={5.2}^{\circ }$ degree to radian

  $\begin{array}{c}\theta ={5.2}^{\circ }\\ =5.2\times 1\text{degree}\\ \left(1\text{degree}=\frac{\pi }{180}\text{radian}\right)\\ \theta =5.2\times \frac{\pi }{180}\text{radian}\\ \theta =0.0907\text{radian}\end{array}$

The distance between the two cities is.

  $s=r\theta$

The radius is $3960\text{miles}$ and the central angle is $\theta =0.0907\text{radian}$

The distance between the two cities is calculated as,

  $\begin{array}{c}s=3960\times 0.0907\\ \text{=359}\text{.17}\text{miles}\end{array}$

Therefore, the distance between two cities Memph is and Louisiana is $s\approx 359\text{miles}$