Starting from one oasis, a camel walks 25 km in a direction 30degrees south of west and then walks 30km toward the north to a second oasis. What distance separates the two oases?

We can solve this problem using vector addition and trigonometry. Here's how:

### Step 1: Breakdown the camel's journey

1. **First leg of the journey**: 
   The camel walks **25 km** in a direction **30° south of west**. 
   - We can break this into two components: west and south.
   - The westward component is:  
     \[
     W_1 = 25 \times \cos(30^\circ)
     \]
   - The southward component is:  
     \[
     S_1 = 25 \times \sin(30^\circ)
     \]

2. **Second leg of the journey**: 
   The camel walks **30 km** directly **north**. This affects only the north-south direction, so it offsets the southward component from the first leg.

### Step 2: Combine the components

Now, let's combine the components:

- **Net westward distance** = \( W_1 \)
- **Net north-south distance**: 
  - Since the camel moved south first and then north, the total north-south displacement is:
    \[
    \text{Net north-south distance} = 30 - S_1
    \]

### Step 3: Calculate the total displacement (distance between the two oases)

The total distance between the two oases is the resultant of these two displacements (westward and north-south). This is given by the Pythagorean theorem:
\[
\text{Total distance} = \sqrt{(W_1)^2 + (\text{Net north-south distance})^2}
\]

Let’s calculate the values and find the final answer.

First, I will compute these values.The distance separating the two oases is approximately **27.84 km**.

Would you like a more detailed explanation of the steps, or do you have any further questions?

Here are five related questions for further exploration:
1. How would the problem change if the camel walked 30° north of west instead of south?
2. What if the camel walked 40 km instead of 30 km north in the second leg?
3. How can you find the angle between the two oases using trigonometry?
4. How would the journey look if plotted on a coordinate grid?
5. What if the camel took three legs instead of two — how would you calculate the total displacement?

**Tip:** Always break down vectors into components when solving problems involving angles and distances!

Starting from one oasis, a camel walks 25 km in a direction 30 degrees south of west and then walks 30 km toward the north to a second oasis. What distance separates the two oases?

Solve a real-world physics problem where a camel walks 25 km in a direction 30 degrees south of west and 30 km toward the north. Learn to use vector addition, trigonometry, and the Pythagorean Theorem to find the distance separating the two oases. Suitable for high school students in Grades 10-12.

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