Question

The figures indicate that the higher the orbit of a satellite, the more of the earth the satellite can "see." Let θ, s, and h be as in the figure, and assume the earth is a sphere of radius 3960 mi.
If the satellite is 90 mi above the earth, what is the distance s that it can see? (Round your answer to the nearest mile.)

How high does the satellite have to be in order to see two cities at the same time if the cities are 3000 mi apart? (Round your answer to the nearest mile.)

Answer 1

For the first problem, we use the trigonometric function cosine. The equation would be:

cos (1/2)θ = 3960 / (3960 + 90)
Solving for θ
θ = 24.20°

Solving for the distance the satellite can see:
s = rθ
s = 3960 (24.20) π/180
s = 1672.79 mi

The satellite can see1672.79 mi of distance.

For the second problem, we approach the reverse manner as the first problem.
Solving for the angle,
θ = s/r
θ = 3000 / 3960
θ = 0.7576 (180/π) = 43.41°

Solving for h:
cos 1/2 (43.41) = 3960 / (3960 + h)
h = 302.19 mi

The satellite must be 302.19 mi above the Earth.