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The problem shown in the image involves navigation, bearings, and trigonometry to calculate the distance and bearing of a boat from a harbor after it has traveled along two different directions.

### Given:
1. **First leg:** The boat travels 30 miles in the direction **N 40° E**.
2. **Second leg:** The boat turns 90° to travel 22 miles in the direction **S 50° E**.

### Approach:
1. **Convert the directions into a coordinate system:**
   - **N 40° E** means the boat is 40° east of due north, so this can be broken into northward and eastward components.
   - **S 50° E** means the boat is 50° east of due south, also broken into components.
   
2. **Calculate the position after each leg of the journey using trigonometry**:
   - Use sine and cosine to break down each leg into north-south and east-west components.

3. **Find the total displacement** from the origin (harbor) using the Pythagorean theorem.

4. **Calculate the bearing** of the boat from the harbor entrance using inverse trigonometry (tan⁻¹).

Let me solve this step by step:

### Step 1: Resolve the first leg (30 miles at N 40° E).
- North component: \( 30 \times \cos(40^\circ) \)
- East component: \( 30 \times \sin(40^\circ) \)

### Step 2: Resolve the second leg (22 miles at S 50° E).
- South component: \( 22 \times \cos(50^\circ) \)
- East component: \( 22 \times \sin(50^\circ) \)

The eastward components from both legs will be added, while the northward and southward components will be subtracted.

### Step 3: Calculate the total displacement and bearing.

I'll proceed to calculate these values.The boat is approximately **37 miles** from the harbor, and the bearing from the harbor is approximately **N 76° E**.

Would you like further details or have any questions?

Here are 5 questions to expand on this problem:
1. How are bearings typically measured in navigation?
2. Why is it important to break down vectors into their components?
3. How does the Pythagorean theorem help in solving displacement problems?
4. What are the common applications of trigonometry in navigation?
5. How do changes in direction affect the final position of a vessel?

**Tip:** When solving navigation problems, always visualize the path and break it into components for accuracy in distance and direction calculations.

A boat leaves the harbor entrance and travels 30 miles in the direction N 40° E. The captain then turns the boat 90° and travels another 22 miles in the direction S 50° E. At that time, how far is the boat from the harbor entrance, and what is the bearing of the boat from the harbor entrance?

Solve a boat navigation problem using trigonometry and bearings. The boat travels 30 miles in the direction N 40° E and then 22 miles S 50° E. Learn how to calculate the total displacement and bearing using the Pythagorean theorem and trigonometric functions. Suitable for Grades 9-12.

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