Chapter 6.8, Problem 13CFU

(a)

To determine

To find the circumference of a great circle.

Answer to Problem 13CFU

  $40,192\text{ km}$ .

Explanation of Solution

Given:

Parallel lines are numbered up to $\frac{\pi }{2}$ the north and to the south. The radius of Earth is about 6400 kilometers. The length of any parallel of latitude is equal to the circumference of a great circle of Earth times the cosine of the latitude angle.

Calculation:

Circumference of a great circle is

  $\begin{array}{l}2\pi R_e\\ =2\times 3.14\times 6400\\ =40,192\text{ km}\end{array}$

(b)

To determine

To write an equation for the circumference of any latitude circle with angle $\theta$ .

Answer to Problem 13CFU

  $2\pi R_e\cos \theta$ .

Explanation of Solution

Given:

Parallel lines are numbered up to $\frac{\pi }{2}$ the north and to the south. The radius of Earth is about 6400 kilometers. The length of any parallel of latitude is equal to the circumference of a great circle of Earth times the cosine of the latitude angle.

Calculation:

Circumference of any latitude circle with angle $\theta$ .

  $2\pi R_e\cos \theta$

(c)

To determine

To find which latitude circle has a circumference of about 3593 kilometers.

Answer to Problem 13CFU

  ${84.89}^{\circ }$ Latitude.

Explanation of Solution

Given:

Parallel lines are numbered up to $\frac{\pi }{2}$ the north and to the south. The radius of Earth is about 6400 kilometers. The length of any parallel of latitude is equal to the circumference of a great circle of Earth times the cosine of the latitude angle.

Calculation:

Circumference of a latitude is

  $\begin{array}{l}2\pi R_e\cos \theta \\ \Rightarrow 40,192\cos \theta =3593\\ \Rightarrow \cos \theta =\frac{3593}{40,192}\\ \Rightarrow \cos \theta =0.089\\ \Rightarrow \theta ={\cos }^{-1}(0.089)\\ \theta ={84.89}^{\circ }\end{array}$

  ${84.89}^{\circ }$ Latitude

(d)

To determine

To find the circumference of the equator.

Answer to Problem 13CFU

  $40,192\text{ km}$ .

Explanation of Solution

Given:

Parallel lines are numbered up to $\frac{\pi }{2}$ the north and to the south. The radius of Earth is about 6400 kilometers. The length of any parallel of latitude is equal to the circumference of a great circle of Earth times the cosine of the latitude angle.

Calculation:

Circumference of equator is

  $\begin{array}{l}2\pi R_e\cos \theta \\ =2\times 3.14\times 6400\cos 0\\ =40,192\times 1\\ =40,192km\end{array}$