Starting from one oasis, a camel walks 25 km in a direction `30^(@)` south of west and then walks 30 km towards the north to a second oasis. What distance separates the two cases?

Options

A. 15 km
B. 48 km
C. 28 km
D. 53 km

Answer

To solve the problem step by step, we will break down the camel's journey into components and then calculate the distance between the two oases.

### Step 1: Understand the Directions and Components
The camel first walks 25 km in a direction that is 30 degrees south of west. We need to resolve this movement into its x (west-east) and y (north-south) components.

### Step 2: Set Up the Coordinate System
- Define the positive x-axis as east and the negative x-axis as west.
- Define the positive y-axis as north and the negative y-axis as south.

### Step 3: Calculate the Components of the First Journey
The camel walks 25 km at an angle of 30 degrees south of west:
- The x-component (west direction) is given by:
\[ x_1 = -25 \cos(30^\circ) \]
- The y-component (south direction) is given by:
\[ y_1 = -25 \sin(30^\circ) \]

Calculating these values:
- \(\cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866\)
- \(\sin(30^\circ) = \frac{1}{2} = 0.5\)

Thus,
\[ x_1 = -25 \times 0.866 \approx -21.65 \text{ km} \]
\[ y_1 = -25 \times 0.5 = -12.5 \text{ km} \]

### Step 4: Calculate the Components of the Second Journey
Next, the camel walks 30 km north. This movement only affects the y-component:
- The x-component remains the same (0 km).
- The y-component changes:
\[ y_2 = 30 \text{ km} \]

### Step 5: Combine the Components
Now, we can find the resultant position of the camel after both journeys:
- The total x-component:
\[ x_{total} = x_1 + 0 = -21.65 \text{ km} \]
- The total y-component:
\[ y_{total} = y_1 + y_2 = -12.5 + 30 = 17.5 \text{ km} \]

### Step 6: Calculate the Distance Between the Two Oases
The distance \(d\) between the two oases can be calculated using the Pythagorean theorem:
\[ d = \sqrt{(x_{total})^2 + (y_{total})^2} \]
Substituting the values:
\[ d = \sqrt{(-21.65)^2 + (17.5)^2} \]
Calculating:
\[ d = \sqrt{469.4225 + 306.25} = \sqrt{775.6725} \approx 27.85 \text{ km} \]

### Final Step: Round the Result
Rounding to the nearest whole number gives us approximately 28 km.

### Conclusion
The distance that separates the two oases is approximately **28 km**.

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