A man walks 700 m due west from A to B. Then he runs 800 m in the direction

S40°W from B to C. Find the true bearing of A from C correct to the nearest degree.

To solve this problem, we need to determine the true bearing of point A from point C after the man has walked from A to B and then from B to C. Here's how to approach it:

### Step 1: Determine the coordinates of points B and C

Let's assume point A is at the origin \((0, 0)\).

- **Point B**: The man walks 700 m due west from A to B. Therefore, the coordinates of point B will be:

  \[
  B(-700, 0)
  \]

- **Point C**: The man then runs 800 m in the direction \( \text{S}40^\circ \text{W} \) from B to C. The direction \( \text{S}40^\circ \text{W} \) means that the angle is measured 40° west of south.

  To find the coordinates of C, we can break down the displacement into its x and y components.

  - The displacement in the x-direction (westward) is:
    \[
    \text{Displacement}_x = 800 \times \sin(40^\circ)
    \]

  - The displacement in the y-direction (southward) is:
    \[
    \text{Displacement}_y = 800 \times \cos(40^\circ)
    \]

  Therefore, the coordinates of point C relative to point B are:
  \[
  C = B + (\text{Displacement}_x, -\text{Displacement}_y)
  \]
  \[
  C = (-700 - 800 \times \sin(40^\circ), 0 - 800 \times \cos(40^\circ))
  \]

### Step 2: Calculate the coordinates of point C

Let's compute the x and y coordinates of C.

\[
\text{Displacement}_x = 800 \times \sin(40^\circ) \approx 800 \times 0.6428 = 514.24 \, \text{m}
\]
\[
\text{Displacement}_y = 800 \times \cos(40^\circ) \approx 800 \times 0.7660 = 612.8 \, \text{m}
\]

So, the coordinates of C are:
\[
C = (-700 - 514.24, 0 - 612.8) = (-1214.24, -612.8)
\]

### Step 3: Find the true bearing of A from C

The bearing is the angle measured clockwise from the north direction to the line joining points A and C. The angle \( \theta \) that we need is the angle between the north direction (positive y-axis) and the line \( \overrightarrow{CA} \).

We can find \( \theta \) using the inverse tangent (arctan) function:

\[
\theta = \arctan\left(\frac{|\text{Change in x}|}{|\text{Change in y}|}\right) = \arctan\left(\frac{|1214.24|}{|612.8|}\right)
\]

\[
\theta = \arctan\left(\frac{1214.24}{612.8}\right) \approx \arctan(1.981) \approx 63.6^\circ
\]

However, this angle \( \theta \) is measured clockwise from the south direction (because we used the absolute values). To find the true bearing from the north:

\[
\text{True Bearing} = 180^\circ + 63.6^\circ = 243.6^\circ
\]

### Final Answer
The true bearing of A from C is approximately **244°** (rounded to the nearest degree).

Would you like more details or have any further questions?

Here are 5 related questions for further practice:
1. What is the distance between points A and C?
2. How would the coordinates of C change if the direction was \( \text{S}50^\circ \text{W} \) instead?
3. Can you find the bearing of C from A?
4. If the man continues to walk 600 m due south from C, what will be his new coordinates?
5. How would you calculate the true bearing if the man had initially walked east instead of west?

**Tip:** Bearings are always measured clockwise from the north direction. Make sure to adjust your angle accordingly based on the quadrant in which your line lies.

Learn how to find the true bearing of point A from point C after a series of displacements. This problem involves trigonometry and vector calculations, covering key concepts such as displacement formulas and true bearings. Suitable for advanced high school students in Grades 10-12.

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