The path of a satellite orbiting the earth causes the satell

The path of a satellite orbiting the earth causes the satellite to pass directly over two tracking stations A and B, which are 65 mi apart. When the satellite is on one side of the two stations, the angles of elevation A and D are measured to be 87.0^+ and 842^+, respectively. (Round your answers to the nearest mile.) (a) How far is the satellite from station A? ms (b) How high the satellite above the ground? ms

Solution

Let the position nof the satellite be denoted by S and let the perpendicular from S to the ground meet the line BA at P. Also, let the satellite be h miles above the ground ans let PA = x miles.

Then in the right angled SPA, we have tan 87⁰ = h/x so that h = x tan 87⁰ = 19.0811x…(1). Similarly, considering the right angled SPB, we have tan 84.2⁰ = h/(x+65) so that h = (x+65) tan 84.2⁰= (x+65)*9.8448 or, h = 9.8448x+ 639.9120...(2). Now, on subtracting the 2^nd equation from the 1^st equation, we get 19.0811x -9.8448x -639.9120 = 0 or, 9.2363x = 639.9120 so that x = 639.9120/9.2363 =69.2823. Then, from the 1^st equation, we get h =19.0811*69.2823 = 1321.9823.Further, in the right angled SPA, we have sin 87⁰ = h/SA so that SA = h sin 87⁰ = 1321.9823*0.9986 = 1320.1706.

Thus, the satellite is 1320.17 miles from station A.

Also, the satellite is 1321.98 miles above the ground.