The path of a satellite orbiting the earth causes it to pass directly over two tracking stations $A$ and $B$, which are 50 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 $87.0°$ and $84.2°$, respectively.
a) How far is the satellite from station A?
b) How high is the satellite above the ground?
So, (a) seemed a pretty straight forward application of the law of sines. I found out that the degree measure between Station A and the ground was $93°$ at which point I applied the law of sines and got:
$$\frac { 50 }{ sin(2.8) } =\frac { d }{ sin(84.2) } $$
$$\frac { (50)sin(84.2) }{ sin(2.8) } =d$$
$$d\approx 1018 \ mi.$$
Now for (b), I am very confused as to how to get this answer… I feel like it would include subtracting the height of the station because the it is getting in the way between the satellite and the ground. But that wouldn't make sense since they didn't give the height of the stations… can someone give me a hint the right direction? No direct answer please
Best Answer
Hint: Draw a vertical line from the satellite $S$ and let $X$ be where this line intersects the ground and let $h$ be the satellite's height. Consider the right triangle $SAX$. Notice that: $$ \sin 87^\circ = \frac{h}{d} $$