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.