Precal Question

Question

Two tracking stations are on the equator 153 miles apart. A weather balloon is located on a bearing of N 40 degrees E from the western station and on a bearing of N 18 degrees E from the eastern station. How far is the balloon from the western station? Round to the nearest mile.

Answer 1

We can solve this problem using the Law of Sines!

Step 1: Understand the Bearings

Let:

W = Western tracking station

E = Eastern tracking station

B = Weather balloon

The two stations are 153 miles apart along the equator.

The balloon is located:

1. N 40° E from the western station.
2. N 18° E from the eastern station.

Remember that bearings are measured from north.

Step 2: Find the Triangle's Angles

At the western station, the angle between east and N 40° E is:

90° - 40° = 50°

Therefore:

Angle W = 50°

At the eastern station, the direction toward the western station is due west.

The angle between west and N 18° E is:

90° + 18° = 108°

Therefore:

Angle E = 108°

The angles in a triangle add up to 180°.

Angle B = 180° - 50° - 108°

Angle B = 22°

Step 3: Apply the Law of Sines

The Law of Sines states:

a/sin(A) = b/sin(B) = c/sin(C)

We know:

Distance WE = 153 miles

Angle B = 22°

Angle E = 108°

We want the distance from the western station to the balloon, WB.

The side WB is opposite angle E.

Therefore:

WB/sin(108°) = 153/sin(22°)

Step 4: Solve for WB

Multiply both sides by sin(108°):

WB = 153 × sin(108°)/sin(22°)

Using a calculator in degree mode:

sin(108°) ≈ 0.95106

sin(22°) ≈ 0.37461

Substitute:

WB ≈ 153 × 0.95106/0.37461

WB ≈ 388.44 miles

Round to the nearest mile:

WB ≈ 388 miles

Final Answer:

The weather balloon is approximately 388 miles from the western tracking station.

Helpful Tip:

When solving bearing problems:

1. Draw a diagram showing north, south, east, and west.
2. Convert the bearings into interior angles of the triangle.
3. Find the missing angle using 180°.
4. Apply the Law of Sines when you know a side and its opposite angle.

The most common mistake is using 40° and 18° directly as the triangle's interior angles. Since bearings are measured from north, you must first convert them to the appropriate angles.