15. A boat travels on a course bearing of $$ S 41^{\prime} 40^{\prime} \mathrm{W} $$ for 84 miles. How far south and how far west is the boat from its starting point?

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To determine how far south and how far west the boat is from its starting point after traveling 84 miles on a bearing of $$ S \, 41^\circ 40' \, W $$, we can break down the displacement into its south and west components using trigonometry.

### Step 1: Understand the Bearing
A bearing of $$ S \, 41^\circ 40' \, W $$ means the boat is traveling:
- **South** by 41°40' towards the **West**.

### Step 2: Convert the Angle to Decimal Degrees
$$ 41^\circ 40' = 41 + \frac{40}{60} = 41.6667^\circ $$

### Step 3: Calculate the South and West Components
Using trigonometric functions:
- **South Component (Adjacent to the angle)**:
  $$
  \text{South} = 84 \times \cos(41.6667^\circ)
  $$
- **West Component (Opposite to the angle)**:
  $$
  \text{West} = 84 \times \sin(41.6667^\circ)
  $$

### Step 4: Compute the Trigonometric Values
Using a calculator:
- $$\cos(41.6667^\circ) \approx 0.7470$$
- $$\sin(41.6667^\circ) \approx 0.6648$$

### Step 5: Calculate the Distances
- **South Distance**:
  $$
  84 \times 0.7470 \approx 62.75 \text{ miles}
  $$
- **West Distance**:
  $$
  84 \times 0.6648 \approx 55.84 \text{ miles}
  $$

### Final Answer
After traveling 84 miles on a bearing of $$ S \, 41^\circ 40' \, W $$, the boat is approximately:

- **62.7 miles south**
- **55.8 miles west**

**Answer:** After traveling 84 mi, the boat is approximately 62.7 miles south and 55.8 miles west of its start.
After traveling 84 miles on a bearing of $$ S \, 41^\circ 40' \, W $$, the boat is approximately 62.7 miles south and 55.8 miles west of its starting point.

15. A boat travels on a course bearing of \( S 41^{\prime} 40^{\prime} \mathrm{W} \) for 84 miles. How far south and how far west is the boat from its starting point?

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