How far is the Satellite?

Question

The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 70 km apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 86.1° and 85.2°, respectively.

How far is the satellite from station A?

distance from A =  km

How high is the satellite above the ground?

height =  km

Answer 1

Point C = Satellite

m∠ACB = 180 - 86.1- 85.2 = 8.7 degrees

(a) We can solve the segment AC using Law of Sine:

sin 8.7° / 70 = sin 85.2° / AC

AC = 70•sin 85.2 / sin 8.7°

AC ≈ 461.15 km is the distance from satellite to Station A

(b) Assuming that segment AB is the ground, then:

sin 86.1º = h / 461.15

h ≈ 460.09 km is the height of the satellite from the ground.

Answer 2

Draw a figure. This problem tacitly assume the earth is flat!

Let C be the point to the left of A where the observations are made.

Angle ABC = 85.2°...because of alternate-interior angles of parallel lines

Angle BCA= 1.3°

Now use the law of sines to get AC.

Then the ht of the satellite is AC sin 86.1°.