A man walks a distance of 18 km in a direction of S24oE.

He then walks 22 km due west. What is the bearing of the man from his starting position?

To determine the bearing of the man from his starting position after walking 18 km in a direction of S24°E and then 22 km due west, we can follow these steps:

1. **Resolve the man's first displacement into components:**

   The man walks 18 km in a direction of S24°E. This means he is 24° east of the south direction. We can break this into southward and eastward components.

   - Southward component: \( 18 \cos(24°) \)
   - Eastward component: \( 18 \sin(24°) \)

   Using the trigonometric values:
   \[
   \cos(24°) \approx 0.9135
   \]
   \[
   \sin(24°) \approx 0.4067
   \]

   - Southward component: \( 18 \times 0.9135 = 16.443 \) km
   - Eastward component: \( 18 \times 0.4067 = 7.321 \) km

2. **Add the second displacement:**

   The man then walks 22 km due west. This affects only the east-west component of his displacement:
   
   - New eastward component: \( 7.321 - 22 = -14.679 \) km (west)
   - The southward component remains \( 16.443 \) km (south)

3. **Calculate the resultant displacement vector:**

   The man's resultant displacement can be represented as a vector with components:
   \[
   \text{Southward} = 16.443 \text{ km}
   \]
   \[
   \text{Westward} = 14.679 \text{ km}
   \]

   We can find the magnitude of the resultant displacement using the Pythagorean theorem:
   \[
   d = \sqrt{(16.443)^2 + (14.679)^2}
   \]
   \[
   d = \sqrt{270.390 + 215.670}
   \]
   \[
   d = \sqrt{486.060} \approx 22.05 \text{ km}
   \]

4. **Calculate the bearing:**

   The bearing is measured clockwise from the north direction. First, find the angle \(\theta\) south of west using the tangent function:
   \[
   \tan(\theta) = \frac{16.443}{14.679}
   \]
   \[
   \theta = \tan^{-1} \left( \frac{16.443}{14.679} \right) \approx \tan^{-1}(1.120) \approx 48.1°
   \]

   Since the direction is south of west, we convert this to a bearing measured clockwise from the north. The bearing is:
   \[
   180° + 48.1° = 228.1°
   \]

So, the bearing of the man from his starting position is approximately \(228.1°\).

Would you like further details or have any questions?

Here are some related questions you might find useful:
1. How do you convert between different bearing notations?
2. What are the steps to resolve a vector into components?
3. How do you use trigonometric functions to find vector components?
4. What is the Pythagorean theorem and how is it used in navigation?
5. How do you determine the direction angle of a resultant vector?
6. How can you find the resultant vector from multiple displacements?
7. How do you interpret and measure bearings on a compass?
8. What are common errors to avoid when calculating bearings?

**Tip:** Always keep track of the direction when resolving vectors into components, as it helps in accurately determining the resultant direction.

Discover how to calculate the bearing of a man from his starting position after walking 18 km in a direction of S24°E and then 22 km due west. This problem involves trigonometric functions, vector components, and the Pythagorean theorem. Suitable for high school students learning trigonometry and vector mathematics.

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