Answer :
Step-by-step explanation:
See image: 'arc' = 104.9° invisible portion = 360 - 104.9 = 255.1°
To determine the measure of the arc that represents the part of Earth the satellite cannot view, we need to consider the geometry of the situation.
Understand the Problem
- The satellite is 1200 miles above the Earth's surface, meaning its total distance from the center of the Earth is the Earth's radius (approximately 4000 miles) plus 1200 miles, which totals 5200 miles.
- The Earth's radius is 4000 miles.
- We need to find the arc of Earth that is not visible to the satellite.
Calculate the Central Angle
- First, calculate the angle at the Earth's center that the radius to the satellite makes with the tangent line at the Earth's surface where the satellite's view begins and ends. This angle is crucial since it helps us define what portion of the Earth's surface is visible.
Apply Geometry to Find the Angle
- Since the satellite's line of sight creates a right triangle with the Earth's radius, we can use the cosine rule to find the angle:
[tex]\cos(\theta) = \frac{\text{Earth's radius}}{\text{distance from Earth's center to the satellite}} = \frac{4000}{5200} = \frac{5}{6.5}[/tex]
- Solving for [tex]\theta[/tex], we find:
[tex]\theta = \cos^{-1}\left(\frac{5}{6.5}\right) \approx 38.21^\circ[/tex]
Calculate the Viewable Arc
- Since the satellite views an arc on both sides (beginning and ending of its view), the total viewable arc is:
[tex]2\theta = 2 \times 38.21^\circ \approx 76.42^\circ[/tex]
Determine the Non-Viewable Arc
- The full circle is 360 degrees. Subtract the viewable arc from this to find the arc the satellite cannot view:
[tex]360^\circ - 76.42^\circ = 283.58^\circ[/tex]
- Therefore, the arc that represents the part of Earth the satellite cannot view is approximately [tex]283.6^\circ[/tex].
Thus, the satellite cannot view an arc of approximately [tex]283.6^\circ[/tex] on Earth.
