Question

the path of a satellite orbiting the earth causes the satellite 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. (round your answers to the nearest mile.) (a) how far is the satellite from station a? (b) how high is the satellite above the ground?

Explanation:

Step1: Find the third - angle of the triangle

The two angles of elevation are $\angle A = 87.0^{\circ}$ and $\angle B=84.2^{\circ}$. The third - angle $\angle C$ of the triangle formed by the two stations and the satellite is $180^{\circ}-(87.0^{\circ}+84.2^{\circ}) = 8.8^{\circ}$. The distance between the two stations $AB = c = 50$ miles.

Step2: Use the Law of Sines to find the distance from station A to the satellite (side b)

By the Law of Sines, $\frac{b}{\sin B}=\frac{c}{\sin C}$. We know that $c = 50$, $\sin B=\sin(84.2^{\circ})\approx0.995$, and $\sin C=\sin(8.8^{\circ})\approx0.153$. Then $b=\frac{c\sin B}{\sin C}=\frac{50\times\sin(84.2^{\circ})}{\sin(8.8^{\circ})}=\frac{50\times0.995}{0.153}\approx325$ miles.

Step3: Find the height of the satellite above the ground (h)

Let the height of the satellite above the ground be $h$. In the right - triangle formed by station A, the point on the ground directly below the satellite, and the satellite, $\sin A=\frac{h}{b}$. Since $A = 87.0^{\circ}$ and $b\approx325$ miles, $h = b\sin A=325\times\sin(87.0^{\circ})\approx325\times0.9986\approx324$ miles.

Answer:

(a) 325 mi
(b) 324 mi