A plane flies 780 miles from Franklin to Centerville with a bearing of 75°. Then it flies 637 miles from Centerville to Rosemount with a bearing of 32°. Draw a diagram that gives a visual representation of the problem. Then find the straight-line distance and bearing from Franklin to Rosemount.

To solve this problem, let's start by analyzing the given data and drawing a diagram.

### Step 1: Draw a Diagram
1. **Franklin to Centerville (First leg):**  
   - Distance: 780 miles  
   - Bearing: 75° (This bearing is measured clockwise from due north.)

2. **Centerville to Rosemount (Second leg):**  
   - Distance: 637 miles  
   - Bearing: 32° (This bearing is also measured clockwise from due north.)

In the diagram:
- Start with a point labeled **Franklin (F)**.
- Draw a line segment from Franklin to Centerville, representing the first leg of the journey. This line should make an angle of 75° with the vertical north direction.
- From Centerville, draw another line segment representing the second leg of the journey to Rosemount. This line should make an angle of 32° with the vertical north direction.

### Step 2: Solve for the Straight-Line Distance and Bearing from Franklin to Rosemount

To find the straight-line distance (d) and bearing (θ) from Franklin to Rosemount:
- We can treat this as a problem of finding the resultant vector in a vector triangle.
- Let \( C = 75^\circ \) and \( B = 32^\circ \) be the bearings, then:
    - The angle between the two legs inside the triangle \( \theta \) is \( 75^\circ - 32^\circ = 43^\circ \).

We will apply the law of cosines to find the distance (d) and then use the law of sines or trigonometric functions to find the bearing.

Let me calculate this for you.### Results:
- The straight-line distance from Franklin to Rosemount is approximately **536.1 miles**.
- The bearing from Franklin to Rosemount is approximately **20.87°**.

### Diagram:
- You can imagine the diagram as follows:
  - **Franklin to Centerville**: A line segment with a length of 780 miles, starting at Franklin and angled 75° from the north.
  - **Centerville to Rosemount**: A second line segment with a length of 637 miles, starting at Centerville and angled 32° from the north.
  - **Franklin to Rosemount**: The straight-line distance and bearing calculated above.

Would you like to see more details or have any questions?

### Related Questions:
1. How does the bearing affect the overall flight path?
2. Can the law of sines be used to find the bearing more easily?
3. What if the angle between the two legs were different, how would that affect the distance?
4. How would the problem change if the plane flew in the opposite direction?
5. Can you explain the law of cosines and how it applies to this problem?

### Tip:
When solving problems involving bearings, always draw a diagram to visualize the situation better. It helps in understanding the angles and applying the correct trigonometric laws.

Learn how to calculate the straight-line distance and bearing from Franklin to Rosemount using trigonometric principles and the law of cosines. This problem involves vector addition and is suitable for high school students.

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