https://file.aiquickdraw.com/mathbot/file/1728588129069v611ae4p.jpg

The diagram you provided shows three tents \( A \), \( B \), and \( C \) arranged in a right-angled triangle. We are tasked with calculating the bearing of point \( C \) from point \( A \). 

The information given is:
- Tent \( B \) is 5 m due south of \( A \) (vertical leg of the triangle).
- Tent \( C \) is 7.5 m due west of \( B \) (horizontal leg of the triangle).

Since the right angle is at \( B \), we need to find the bearing of \( C \) from \( A \). Bearings are measured clockwise from the north direction. In this case, we need to find the angle \( \theta \) that tent \( C \) makes with respect to the north direction from \( A \).

### Steps to Solve:
1. **Determine the angle \( \theta \)** using trigonometry:
   The triangle formed by \( A \), \( B \), and \( C \) is a right triangle. The angle \( \theta \) is the angle between the line \( AC \) and the vertical leg \( AB \). We can use the tangent function:
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{7.5}{5}
   \]
   So, 
   \[
   \theta = \tan^{-1}\left(\frac{7.5}{5}\right)
   \]

2. **Calculate the value of \( \theta \)**:
   \[
   \theta = \tan^{-1}(1.5) \approx 56.31^\circ
   \]

3. **Calculate the bearing**:
   Bearings are measured clockwise from the north, so we need to add \( 180^\circ \) to the calculated angle \( \theta \), because the direction of \( C \) from \( A \) is in the southwest quadrant. Thus, the bearing of \( C \) from \( A \) is:
   \[
   180^\circ + 56.31^\circ = 236.3^\circ
   \]

### Final Answer:
The bearing of \( C \) from \( A \) is approximately \( 236.3^\circ \).

---

Would you like further details or explanations? 

Here are 5 related questions to explore:
1. How do you calculate a bearing if you are given coordinates instead of distances?
2. What other trigonometric functions could be useful in finding angles in triangles?
3. How would you calculate the distance between \( A \) and \( C \)?
4. What is the difference between true bearing and relative bearing?
5. Can bearings ever be negative, and if so, how are they handled?

**Tip**: Always ensure you know whether bearings are being measured clockwise or counterclockwise, as this can impact the final answer significantly.

Calculate the bearing of C from A given the positions of three tents, A, B, and C. B is 5 m due south of tent A. C is 7.5 m due west of tent B.

Learn how to calculate the bearing of C from A in a right-angle triangle formed by tents A, B, and C. Use trigonometry to determine the angle and find the final bearing in a detailed step-by-step solution. Ideal for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation