What is the significance of change in angle from 60 to 45 in this problem

The change in the angle of elevation from \( 60^\circ \) to \( 45^\circ \) in this problem is significant because it directly affects the horizontal distance between the boat and the base of the lighthouse. Here's why:

### 1. **Relationship Between Angle of Elevation and Distance:**
   - The angle of elevation is the angle formed by the line of sight from an observer (in this case, at the boat) to the top of an object (the lighthouse) and the horizontal line from the observer to the base of the object.
   - When the boat moves away from the lighthouse, the angle of elevation decreases because the distance between the boat and the lighthouse increases.
   - The angles \( 60^\circ \) and \( 45^\circ \) correspond to specific horizontal distances from the base of the lighthouse. Specifically:
     - At \( 60^\circ \), the horizontal distance \( d_1 \) is smaller.
     - At \( 45^\circ \), the horizontal distance \( d_2 \) is larger.

### 2. **Mathematical Implication:**
   - The tangent of the angle of elevation is the ratio of the height of the lighthouse to the horizontal distance from the boat to the lighthouse:
     \[
     \tan(\theta) = \frac{\text{height of the lighthouse}}{\text{horizontal distance from lighthouse}}
     \]
   - For \( \theta = 60^\circ \), \( \tan(60^\circ) = \sqrt{3} \), meaning the horizontal distance is less.
   - For \( \theta = 45^\circ \), \( \tan(45^\circ) = 1 \), meaning the horizontal distance is equal to the height of the lighthouse.

### 3. **Physical Interpretation:**
   - As the angle of elevation changes from \( 60^\circ \) to \( 45^\circ \), the boat has moved farther away from the lighthouse.
   - This change in angle allows us to calculate the distance the boat has traveled during this time interval, which is then used to determine the boat's speed.

### 4. **Key Takeaway:**
   - The change in angle from \( 60^\circ \) to \( 45^\circ \) provides the necessary information to compute how far the boat has traveled horizontally. This information is crucial for calculating the boat's speed using the time it took for this change to occur.

Would you like to explore more details or have any additional questions?

Here are 8 related questions:
1. How would the calculation differ if the angles were \( 45^\circ \) and \( 30^\circ \) instead?
2. What is the relationship between speed and angle of elevation in this type of problem?
3. Can you explain how trigonometric functions like sine or cosine might be used in similar problems?
4. What other factors could influence the angle of elevation in real-world scenarios?
5. How does the height of the lighthouse affect the angles of elevation for the same distances?
6. What would happen to the speed calculation if the boat took longer to change the angle from \( 60^\circ \) to \( 45^\circ \)?
7. How might wind or water currents affect the boat's speed and the angle change?
8. Could this problem be solved using a different trigonometric approach, such as using the sine or cosine laws?

**Tip:** When solving trigonometric problems, sketching the scenario can provide valuable insights and help visualize the relationships between angles and distances.

Understand the significance of the angle change from 60 degrees to 45 degrees in elevation problems. Explore how this change impacts distance calculations and the relationship between angle of elevation and horizontal distance. Suitable for high school students studying trigonometry and related concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation