Chapter 6.1, Problem 56E

a.

To determine

To find: approximate distance between Minneapolis and Arkadelphia.

Answer to Problem 56E

  $760.3\text{ miles}$

Explanation of Solution

Given information:

The latitude of Minneapolis = $45°\text{N}$

The latitude of Arkadelphia = $34°\text{N}$

The latitude of Alexandria = $31°\text{N}$

Radius of earth $\left(r\right)$ = $3960\text{ miles}$

Minneapolis, Arkadelphia, and Alexandria lie on same longitude line.

Calculation:

Look at the figure to understand the angle between the Minneapolis and Arkadelphia.

  

The angle between Minneapolis and Arkadelphia is evaluated as follows:

  $\begin{array}{c}\theta =45°-34°\\ =11°\\ =11°\left(\frac{\pi }{180°}\right)\left[\text{Convert into radian measure}\right]\\ =\frac{11\pi }{180}\text{ radians}\end{array}$

The approximate distance between Minneapolis and Arkadelphia will be the arc length of the sector as shown above. The value of $s$ can be evaluated as follows:

  $\begin{array}{c}s=r\theta \\ =\left(3960\right)\left(\frac{11\pi }{180}\right)\left[\text{Put the value of }r\text{ and }\theta \right]\\ =760.3\text{ miles}\end{array}$

Thus the approximate distance between Minneapolis and Arkadelphia is $760.3\text{ miles}$ .

b.

To determine

To find: approximate distance between Minneapolis and Alexandria.

Answer to Problem 56E

  $967.6\text{ miles}$

Explanation of Solution

Given information:

The latitude of Minneapolis = $45°\text{N}$

The latitude of Arkadelphia = $34°\text{N}$

The latitude of Alexandria = $31°\text{N}$

Radius of earth $\left(r\right)$ = $3960\text{ miles}$

Minneapolis, Arkadelphia, and Alexandria lie on same longitude line.

Calculation:

Look at the figure to understand the angle between the Minneapolis and Alexandria.

  

The angle between Minneapolis and Alexandria is evaluated as follows:

  $\begin{array}{c}\theta =45°-31°\\ =14°\\ =14°\left(\frac{\pi }{180°}\right)\left[\text{Convert into radian measure}\right]\\ =\frac{7\pi }{90}\text{ radians}\end{array}$

The approximate distance between Minneapolis and Alexandria will be the arc length of the sector as shown above. The value of $s$ can be evaluated as follows:

  $\begin{array}{c}s=r\theta \\ =\left(3960\right)\left(\frac{7\pi }{90}\right)\left[\text{Put the value of }r\text{ and }\theta \right]\\ =967.6\text{ miles}\end{array}$

Thus the approximate distance between Minneapolis and Alexandria is $967.6\text{ miles}$ .

c.

To determine

To find: approximate distance between Arkadelphia and Alexandria.

Answer to Problem 56E

  $207.3\text{ miles}$

Explanation of Solution

Given information:

The latitude of Minneapolis = $45°\text{N}$

The latitude of Arkadelphia = $34°\text{N}$

The latitude of Alexandria = $31°\text{N}$

Radius of earth $\left(r\right)$ = $3960\text{ miles}$

Minneapolis, Arkadelphia, and Alexandria lie on same longitude line.

Calculation:

Look at the figure to understand the angle between the Arkadelphia and Alexandria.

  

The angle between Arkadelphia and Alexandria is evaluated as follows:

  $\begin{array}{c}\theta =34°-31°\\ =3°\\ =3°\left(\frac{\pi }{180°}\right)\left[\text{Convert into radian measure}\right]\\ =\frac{\pi }{60}\text{ radians}\end{array}$

The approximate distance between Arkadelphia and Alexandria will be the arc length of the sector as shown above. The value of $s$ can be evaluated as follows:

  $\begin{array}{c}s=r\theta \\ =\left(3960\right)\left(\frac{\pi }{60}\right)\left[\text{Put the value of }r\text{ and }\theta \right]\\ =207.3\text{ miles}\end{array}$

Thus the approximate distance between Arkadelphia and Alexandria is $207.3\text{ miles}$ .