To find the bearing of the man from his starting position after walking 12 km in a direction of S37°E and then 25 km due west, we will use vector addition and trigonometry.

Step-by-Step Solution:

1. Break down the distances into their components:

   • For the 12 km walk in the direction of S37°E:

     • South component: $12 \sin(37°)$
     • East component: $12 \cos(37°)$

   • For the 25 km walk due west:

     • West component: 25 km
     • There is no north/south component.

2. Calculate the components:

   • $12 \sin(37°) = 12 \times 0.6018 = 7.2216$ km (South)
   • $12 \cos(37°) = 12 \times 0.7986 = 9.5832$ km (East)

3. Combine the components:

   • South component: 7.2216 km (from the first leg)
   • East component: 9.5832 km (from the first leg)
   • West component: 25 km (from the second leg)

   Since east and west are opposite directions, we subtract the east component from the west component to get the resultant westward distance:

   • Net east/west component: $25 - 9.5832 = 15.4168$ km west

   The net south component remains as 7.2216 km.

4. Determine the resultant displacement vector:

   • South component (y): 7.2216 km
   • West component (x): 15.4168 km

5. Calculate the magnitude of the resultant vector:

   $\text{Resultant distance} = \sqrt{(15.4168^2 + 7.2216^2)}$

   $\text{Resultant distance} = \sqrt{(237.6865 + 52.1432)} = \sqrt{289.8297} = 17.02 \text{ km}$

6. Calculate the bearing angle (θ) from the starting position:

   The bearing angle is measured clockwise from north. We first find the angle from the west direction using arctan:

   $\theta = \tan^{-1}\left(\frac{\text{south component}}{\text{west component}}\right) = \tan^{-1}\left(\frac{7.2216}{15.4168}\right)$

   $\theta = \tan^{-1}(0.4685) = 25.09^\circ$

   Since the direction is south-west, we measure the bearing from north:

   $\text{Bearing} = 180^\circ + 25.09^\circ = 205^\circ \text{ (nearest degree: 205°)}$

Final Answer:

The bearing of the man from his starting position to the nearest degree is 205°.

Would you like more details or have any questions?

Related Questions:

1. How do you convert between different coordinate systems?
2. What is the significance of bearings in navigation?
3. How do you solve similar problems using the Law of Cosines?
4. Can you explain the concept of vector addition in detail?
5. How do you find the resultant vector for non-perpendicular components?
6. What are some real-life applications of vector components?
7. How can trigonometry be used to solve navigation problems?
8. What are some common mistakes to avoid in vector addition problems?

Tip:

Always double-check the direction and angle measurements when working with bearings to ensure accuracy.