Chapter 12.8, Problem 9P

To determine

Find the greatest latitude from the satellite to the Earth.

Answer to Problem 9P

  $\theta \approx 81°$

The length of greatest latitude 42682.55 km.

Explanation of Solution

Given:

A communication satellite is in orbit 35,800 km above the equator.It completes one orbit every 24 hours , so that from Earth it appears to be stationary above a point on the equator. This point has the same longitude as Houston. The satellite’s angle of elevation from Houston.The latitude of Houston is $29.7°N$ .The radius of Earth is 6400 km.

Calculation:

Represent the situation :

  

The distance between Centre of Earth and Houston is 6400 km .

The distance between Satellite and Centre of Earth is 35,800 + 6400 = 42200 km

The greatest latitude it that which is tangent from satellite to the surface of Earth.

We need to find $\theta$ :

  $\begin{array}{l}\cos \theta =\frac{\text{length of the side adjacent to }\theta }{\text{length of hypotenuse}}\\ \Rightarrow \cos \theta =\frac{6400}{42200}\\ \Rightarrow \cos \theta \approx 0.1517\\ \Rightarrow \theta \approx {\cos }^{-1}(0.1517)\mathrm{.............}[{\text{take cos}}^{-1}\text{ on each side]}\\ \Rightarrow \theta \approx \text{81}°\end{array}$

Now, find the length of the latitude by Pythagoras Theorem:

  $\begin{array}{l}x=\sqrt{{6400}^2+{42200}^2}\\ \Rightarrow x=\sqrt{40960000+1780840000}\\ \Rightarrow x\approx 42682.55\end{array}$

So, the length of greatest latitude 42682.55 km.