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=(adjacent)/(hypotenus) \\ cos\theta=(6387)/(30250) \end{gathered}`
`\begin{gathered} \theta=\cos_^(-1)((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=(\alpha)/(360)*2\pi R \\ L=(155.6216068)/(360)*3*\pi*6387 \end{gathered}`
`\begin{gathered} L=26021.68635km \\ The\text{ circumference of the earth=2}*\pi*6387=\:40130.70455km \\ Percentage\text{ of the earth/s circumference within satellite's transmisson is;} \\ =(26021.68635)/(40130.70455)*100=64.84233\text{ \%} \end{gathered}`

Answer;

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