Question

Determine the percentage of the Earth's circumference that lies within a satellite's transmission range if the satellite is 30 250 km above Earth's surface. Note: Earth's radius is approximately 6387 km. (adapted from Big Ideas from Dr. Small Grade 9-12)

Answer 1

Step 1

Given;

$\begin{gathered} D=30250km \\ R=6387km \end{gathered}$

Step 2

$\begin{gathered} Find\text{ the central angle using the trigonometric ratio Cah} \\ \end{gathered}$$\begin{gathered} cos\theta=\frac{adjacent}{hypotenus} \\ cos\theta=\frac{6387}{30250} \end{gathered}$$\begin{gathered} \theta=\cos_^{-1}(\frac{6387}{30350}) \\ \theta=77.81080342^{\circ} \\ central\text{ angle=2}\theta=155.6216068^o \end{gathered}$

Step 3

Determine the percentage of the Earth's circumference that lies within a satellite's transmission range using the length of an arc

$\begin{gathered} L=\frac{\alpha}{360}\times2\pi R \\ L=\frac{155.6216068}{360}\times3\times\pi\times6387 \end{gathered}$$\begin{gathered} L=26021.68635km \\ The\text{ circumference of the earth=2}\times\pi\times6387=\:40130.70455km \\ Percentage\text{ of the earth/s circumference within satellite's transmisson is;} \\ =\frac{26021.68635}{40130.70455}\times100=64.84233\text{ \%} \end{gathered}$

Answer;

$\begin{gathered} =64.84233\% \\ \end{gathered}$