Answer
(a) 1018 mi
(b) 1017 mi
Work Step by Step
(a)
The triangle formed between the two red lines has:
$\angle B = 84.2 ^o$
$\angle A = 180^o - 87.0 ^o = 93^o$
1. Calculate $\angle C$
$$\angle A + \angle B + \angle C = 180^o$$ $$\angle C = 180^o - \angle A - \angle B$$ $$\angle C = 180^o - 93^o - 84.2^o = 2.8 ^o$$
2. Use the law of sines to calculate $c$:
$$\frac{sin \space C}{c} = \frac{sin \space B}{b}$$ $$b \space sin \space C= c \space sin \space B $$ $$b = c \space \frac{sin \space B}{sin \space C} = (50 \space mi)\frac{sin \space 84.2^o}{sin \space 2.8^o} = 1018 \space mi$$
(b)
To calculate the height of the satellite, we can imagine a right triangle that has $\angle A$ as one of its angles, and $a$ is the satellite's height. If $C$ is the 90$^o$ angle, then $c = 1018 \space mi$, as we have calculated on (a), which is the distance between the satellite and station A.
- Using the law of sines:
$$a = c \space \frac{sin \space A}{sin \space C} = (1018 \space mi)\frac{sin \space 87.0 ^o}{sin \space 90^o} = 1017 \space mi$$
