Chapter 12.7, Problem 21PPS

a.

To determine

Find the distance between Phoenix and Helena. Explain.

Answer to Problem 21PPS

The distance will be $913mi$ .

Explanation of Solution

Given:

It is given in the question that the location Phoenix , Arizona is ${112}^{\circ }$ W longitude, ${33.4}^{\circ }$ N latitude and the location of Helena, Montana, is ${112}^{\circ }$ W longitude, ${46.6}^{\circ }$ N latitude west indicates the location in terms of the prime meridian, and north indicates the location. The mean radius of Earth is $3963$ miles.

Concept Used:

In this, use the concept of using the formula of arc length i.e. $s=\frac{\theta }{360}\cdot 2\pi r$ .

Calculation:

Here, the first method is to find arc length between the two cities.

Since, the two cities have the same longitude so they will lie on the same great circle.

  ${40.6}^{\circ }-{33.4}^{\circ }={13.2}^{\circ }$

Now, using the formula of arc length,

  $\begin{array}{l}s=\frac{\theta }{360}\cdot 2\pi r\\ s=\frac{13.2}{360}\cdot 2\pi (3963)\\ s=\frac{328516.8}{360}\\ s\approx 913mi\end{array}$

Conclusion:

The distance will be $913mi$ .

b.

To determine

Explain whether there will any other way to express the distance between these cities.

Answer to Problem 21PPS

It is will done by using the cosine by taking it as a triangle if it were.

Explanation of Solution

Given:

It is given in the question that the location Phoenix , Arizona is ${112}^{\circ }$ W longitude, ${33.4}^{\circ }$ N latitude and the location of Helena, Montana, is ${112}^{\circ }$ W longitude, ${46.6}^{\circ }$ N latitude west indicates the location in terms of the prime meridian, and north indicates the location. The mean radius of Earth is $3963$ miles

Here, if let another method would be to use the idea of a triangle formed by the radii of the earth and the distance between the two cities, then using the law of cosines to find the distance between the two cities. This will give the distance between the cities if it were not on a circle.

c.

To determine

Explain whether the distance between Washington, D.C. and London, England which lie on the approximately the same lines of latitude be calculated in the same way.

Answer to Problem 21PPS

It would be impossible to find the arc length so as to get the distance.

Explanation of Solution

Given:

It is given in the question that the location Phoenix , Arizona is ${112}^{\circ }$ W longitude, ${33.4}^{\circ }$ N latitude and the location of Helena, Montana, is ${112}^{\circ }$ W longitude, ${46.6}^{\circ }$ N latitude west indicates the location in terms of the prime meridian, and north indicates the location. The mean radius of Earth is $3963$ miles

Since, the lines of the latitude are not great circles. There would be a great circle through the points but the arc measures would be unknown. This would make it impossible to find arc length.

d.

To determine

Find the other location that have the same distance from Phoenix, Arizona as Helena.

Answer to Problem 21PPS

There would be infinitely many location that have the same distance.

Explanation of Solution

Given:

It is given in the question that the location Phoenix , Arizona is ${112}^{\circ }$ W longitude, ${33.4}^{\circ }$ N latitude and the location of Helena, Montana, is ${112}^{\circ }$ W longitude, ${46.6}^{\circ }$ N latitude west indicates the location in terms of the prime meridian, and north indicates the location. The mean radius of Earth is $3963$ miles.

Conclusion:

Since, there would be an infinite number of points that are the same distance away from Phoenix. These points would all be in a circle with Phoenix as the center and a Helena as one of the points on the circle.