The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B

Question

The path of a satellite orbiting the earth causes it to pass directly over two tracking stations A and B, which are 57 mi apart. When the satellite is on one side of the two stations, the angles of elevation at A and B are measured to be 87.5° and 84.5°, respectively.

How far is the satellite from station A?

distance from A = ____ mi

How high is the satellite above the ground?

height =  _____ mi

Enter your answer as a number; your answer should be accurate to 2 decimal places.

Answer 1

Draw line AB. Mark AB as 57 mi.

Plot S (for satellite above and to the left of A

Draw SA and SB

Plot point G perpendicular from S to the ground

Mark angle SAG 87.5° angle SB 84.5°

Neither distance can be determine directly.

tan 84.5° = SG / (GA + 57)

tan 84.5° (GA + 57) = SG

GA tan 84.5° + 57 tan 84.5° = SG

GA tan 84.5° = SG - 57 tan 84.5°

GA = (SG - 57 tan 84.5°) / tan 84.5° (EQ. 1)

One more

tan 87.5° = SG / GA

GA tan 87.5° = SG

GA = SG / tan 87.5° (EQ. 2)

Set EQ1 = EQ2 to solve for SG, this is answer to second part.

sin 87.5° = SG / SA, substitute SG and solve for SA, answer to first part.