two hikers set out from the same campsite.one walks 7km in the direction 043 degrees and the other walks 10km in the direction 133 degrees.what is the distance between the two hikers

To find the distance between the two hikers who set out from the same campsite in different directions, we can use the Law of Cosines. This approach is effective because we have two sides of a triangle (the distances each hiker walked) and the angle between their paths.

### Given:
- Hiker 1 walks 7 km in the direction of 043°.
- Hiker 2 walks 10 km in the direction of 133°.
- The angle between their paths: \(\theta = 133^\circ - 43^\circ = 90^\circ\)

### Step 1: Identify the known values
- Side \( a = 7 \) km
- Side \( b = 10 \) km
- Angle between them \( \theta = 90^\circ \)

### Step 2: Apply the Law of Cosines
The Law of Cosines formula is:

\[
c = \sqrt{a^2 + b^2 - 2ab \cdot \cos(\theta)}
\]

Substituting the known values:

\[
c = \sqrt{7^2 + 10^2 - 2 \cdot 7 \cdot 10 \cdot \cos(90^\circ)}
\]

### Step 3: Simplify
Since \(\cos(90^\circ) = 0\), the formula reduces to:

\[
c = \sqrt{7^2 + 10^2}
\]

\[
c = \sqrt{49 + 100} = \sqrt{149}
\]

\[
c \approx 12.21 \text{ km}
\]

### Conclusion
The distance between the two hikers is approximately **12.21 km**.

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Would you like further details, or do you have any other questions? Here are some related questions you might find useful:

1. How would the distance change if the angle between the hikers' paths was different?
2. How can you find the bearing of one hiker from the other using the distance and directions?
3. What other trigonometric methods can be used to find distances in navigation?
4. How does the Law of Cosines relate to the Pythagorean theorem?
5. Can you calculate the exact position (coordinates) of each hiker after walking a certain distance?

**Tip:** In navigation, it's crucial to pay attention to the angles you use. Ensure they are correctly interpreted, especially when dealing with bearings and directions!

Learn how to calculate the distance between two hikers who set out from the same campsite in different directions using the Law of Cosines. This problem involves trigonometric calculations with given distances and angles. Suitable for Grades 10-12.

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