Question

Two weather tracking stations are on the equator 159 miles apart. A weather balloon is located on a bearing of N 38°E from the western station and on a bearing of N 14°E from the eastern station. How far is the balloon from the western station?

Answer 1

First we need to figure out the angles from the equator instead of from N.

We will use A for the west station and B for the east station:
\[A = 90 - 38 = 52^{\circ}\]
since the balloon is east from the eastern station, we need to add 90 degrees for angle B instead of subtract.
\[B = 90 + 14 = 104^{\circ}\]
The last angle is found because a A + B + C = 180
\[C = 180 - A - B = 24^{\circ} \]
now we will set c = 159
and the sine rule comes into play:
\[\frac{a}{sin A} = \frac{b}{sinB} = \frac{c}{sinC}\]
\[\frac{159}{sin(24)} = \frac{a}{sin(52)}\]
\[a = \frac{159*sin(52)}{sin(24)} \approx 308.0463\]
\[b = \frac{159*sin(104)}{sin(24)} \approx 379.3045\]

Where a is the distance from the Eastern station and b is the distance from the western station.