Satellite Trigonometry

A surveillance satellite circles Earth at a height of h kilometers above the surface. Suppose that d is the distance, in kilometers, on the surface of Earth that can be observed from the satellite. The `sight-line` is limited by the horizon, i.e. tangent to the Earth (there is your right angle). Assume Earth radius r=6371 km.

Q1: Find an equation that relates the central angle A (the angle between the two points on the surface of Earth where the sight-line is tangent to the surface) to the height h. A: cos(A/2) = 6371/(6371+h)

Q2: Find an equation that relates the observable distance d and A. A: d = 6371A

Q3: Find an equation that relates d and h. A: cos(d/12742) = 6371/(6371+h)

Q4: If d is to be 4000 km, how high must the satellite orbit above Earth? A: 327.35 km

Q5: If the satellite orbits at height h=500 km, what distance d on the surface can be observed? A: 4891 km