Chapter 4.8, Problem 50E

Solution

To find: The arc length of coverage area for the cable television satellite on Earth’s surface.

The arc length of coverage area for the cable television satellite on Earth’s surface is approximately 1770.46 mi

Given information:

Many satellites travel in geosynchronous orbits, which means that the satellite stays over the same point on our planet. A satellite that broadcasts cable television is in geosynchronous orbit 100 mi above Earth. The Earth is a sphere with radius 4000 mi.

Calculation:

The figure representing the data is given by:

  

The line joining the center of the earth and the satellite is given by:

  $\begin{array}{c}OS=4000+100\\ =4100\end{array}$

The triangles OAS and OBS are congruent.

Consider the triangle OAS,

Using the definitions of trigonometric identities:

  $\begin{array}{c}\cos \theta =\frac{\text{adjacent side}}{\text{hypotenuse}}\\ \cos \left(AOS\right)=\frac{4000}{4100}\\ \angle {\text{AOS=cos}}^{-1}\left(\frac{40}{41}\right)\\ \angle AOS\approx 12.68°\end{array}$

Hence,

  $\begin{array}{c}\angle AOB=2\angle AOS\\ =2\left(12.68°\right)\\ =25.36°\end{array}$

The arc length of coverage area is given by:

  $\begin{array}{l}L=\frac{x}{360°}\left(2\pi r\right)\\ L=\frac{25.36°}{360°}\left(2\pi \cdot 4000\right)\\ L\approx 1770.46\end{array}$

The arc length of coverage area for the cable television satellite on Earth’s surface is approximately 1770.46 mi

Conclusion:

The arc length of coverage area for the cable television satellite on Earth’s surface is approximately 1770.46 mi