Two boats depart from the same dock in a marina.  A sailboat sails at 9 km/h in a direction of 12o west of north.  At the same time, a power bot travels at 15 km/h in a direction of 16o east of north.  How far apart are the boats after 3 hours? Draw a picture to set up this problem.

To determine how far apart the two boats are after 3 hours, we can use the law of cosines, given that the two boats are moving in slightly different directions but with a common starting point.

### Step-by-Step Solution

1. **Determine the distances traveled by each boat** after 3 hours:
   - Sailboat: \( 9 \, \text{km/h} \times 3 \, \text{hours} = 27 \, \text{km} \)
   - Powerboat: \( 15 \, \text{km/h} \times 3 \, \text{hours} = 45 \, \text{km} \)

2. **Set up the angle between their directions**:
   - The sailboat is moving at \(12^\circ\) west of north, while the powerboat is moving at \(16^\circ\) east of north. Therefore, the angle between their directions is:
     \[
     12^\circ + 16^\circ = 28^\circ
     \]

3. **Draw a Diagram**:
   Draw two vectors from a common point:
   - A vector representing the sailboat's path (27 km at \(12^\circ\) west of north).
   - A vector representing the powerboat's path (45 km at \(16^\circ\) east of north).
   - The angle between these two vectors is \(28^\circ\).

4. **Apply the Law of Cosines** to find the distance \(d\) between the boats:
   \[
   d^2 = 27^2 + 45^2 - 2 \cdot 27 \cdot 45 \cdot \cos(28^\circ)
   \]

5. **Calculate \(d\)**:
   \[
   d = \sqrt{27^2 + 45^2 - 2 \cdot 27 \cdot 45 \cdot \cos(28^\circ)}
   \]
   - \(d = \sqrt{729 + 2025 - 2430 \cdot \cos(28^\circ)}\)

Let's proceed with this calculation:

### Final Calculation
Using \(\cos(28^\circ) \approx 0.8829\):
\[
d = \sqrt{729 + 2025 - 2430 \times 0.8829}
\]
\[
d = \sqrt{729 + 2025 - 2145.447} = \sqrt{608.553} \approx 24.66 \, \text{km}
\]

### Answer:
After 3 hours, the boats are approximately **24.66 km apart**.

Would you like more details, or have any questions? 

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Here are five additional questions you might explore related to this setup:

1. How would the solution change if the boats had different speeds?
2. What if both boats traveled at the same speed; would their separation distance change?
3. How could this scenario change if one boat traveled southward?
4. How would the calculation adjust if the boats departed at different times?
5. What if the boats changed direction midway through their journey?

**Tip:** In problems involving multiple directions and distances, sketching a diagram and identifying angles is a useful strategy to clarify vector relationships.

Two boats depart from the same dock in a marina. A sailboat sails at 9 km/h in a direction of 12° west of north. At the same time, a powerboat travels at 15 km/h in a direction of 16° east of north. How far apart are the boats after 3 hours?

Explore how to calculate the distance between two boats after 3 hours, given their speeds and directions. Learn how to use the Law of Cosines for vector-based distance calculations with step-by-step guidance. Ideal for students in Grades 10-12 studying trigonometry and vector analysis.

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