Solve the word problem

Question

Two tracking stations are on the equator 121 miles apart. A weather balloon is located on a bearing of N 38degrees E from the western station and on a bearing oF N 22degreesE from the eastern station. how far is the balloon from the western station?

Answer 1

In this problem it helps to draw a picture. In my labeling convention, I labeled the western station P and the eastern station Q, with the weather balloon labeled B. Then I drew a triangle connecting all three points.

We know one side of the triangle, but none of the angles — however, we can find them based on the bearings given for both tracking stations. The bearing of N 38°E from P to B means a person at P can turn an additional 52° toward the east to look at Q. Similarly, the bearing of of N 22°E from Q to B means a person at Q has to turn westward a total of 112° from the balloon to look at P.

So our angles in the triangle are as follows:

Angle P is 52°

Angle Q is 112°

Angle B is 16° (since all three angles add to 180°)

With one side given and all three angles calculated, we can use the Law of Sines to calculate the distance PB. The 121-mile distance for PQ is opposite the 16° angle B, and our unknown PB is opposite the 112° angle Q. Therefore:

121 / sin(16°) = (PB) / sin(112°)

And so

PB = 121 • sin(112°) / sin(16°)

To the nearest mile, this means the weather balloon is about 407 miles away from the western station.