Athish and Vishnu are part of a scientific team studying thunderclouds. The team is about to
launch a weather balloon into an active part of a cloud. Vishnu’s rope is 6.2 m long and makes
an angle of 360 with the ground. Athish’s rope is 4.9 m long. How far, to the nearest tenth of a
metre, is Vishnu from Athish? SHOW ALL POSSIBILITIES with a diagram, and round any angles
used to the nearest DEGREE.

To solve this problem, we need to analyze the situation where Athish and Vishnu are holding ropes at different angles and lengths. The ropes form two sides of a triangle, and the distance between them is the third side of the triangle, which we need to find.

### Given:
- Vishnu's rope: \( 6.2 \) m
- Angle between Vishnu's rope and the ground: \( 360^\circ \) (Note: This is likely a typo because an angle of \( 360^\circ \) would bring the rope back to the starting position. It should be \( 0^\circ \) or some angle close to that. For calculations, I'll assume it's \( 0^\circ \).)
- Athish's rope: \( 4.9 \) m

We need to determine the distance between the points where Vishnu and Athish are holding the ropes, which we will call \( d \).

### Step 1: Assumptions about the Angle
Given the assumption of \( 0^\circ \) for Vishnu's angle, the situation becomes straightforward if we consider that Vishnu's rope is along the x-axis (horizontal). If we also assume that Athish's rope is at some angle \( \theta \) with respect to the ground, then we can use the Law of Cosines to find the distance between Vishnu and Athish.

### Step 2: Use the Law of Cosines
If \( \theta \) is the angle between the two ropes (angle between Vishnu's rope and Athish's rope), then the distance \( d \) between them is given by:
\[
d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(\theta)}
\]

Given that \( \theta = 0^\circ \) (Vishnu's rope is horizontal, and we consider Athish’s rope to be vertical, so \( \theta = 90^\circ \)), the formula simplifies, but let's consider the other possibilities:

1. **Case 1**: \( \theta = 0^\circ \)
    \[
    d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(0^\circ)}
    \]
    \[
    \cos(0^\circ) = 1
    \]
    \[
    d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times 1}
    \]
    \[
    d = \sqrt{38.44 + 24.01 - 60.76} \approx \sqrt{1.69} \approx 1.3 \text{ meters}
    \]

2. **Case 2**: \( \theta = 90^\circ \)
    \[
    d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(90^\circ)}
    \]
    \[
    \cos(90^\circ) = 0
    \]
    \[
    d = \sqrt{(6.2)^2 + (4.9)^2}
    \]
    \[
    d = \sqrt{38.44 + 24.01} \approx \sqrt{62.45} \approx 7.9 \text{ meters}
    \]

3. **Case 3**: \( \theta = 180^\circ \)
    \[
    d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(180^\circ)}
    \]
    \[
    \cos(180^\circ) = -1
    \]
    \[
    d = \sqrt{38.44 + 24.01 + 60.76} \approx \sqrt{123.21} \approx 11.1 \text{ meters}
    \]

### Conclusion:
- The distance between Vishnu and Athish ranges from \( 1.3 \) meters to \( 11.1 \) meters depending on the angle between the ropes.

### Diagram:
Here’s how you can visualize it:

1. **Case 1**: Both ropes are along the same line (horizontal).
2. **Case 2**: The ropes are perpendicular to each other.
3. **Case 3**: The ropes are in opposite directions.

### To Summarize:
The distance between Vishnu and Athish can vary depending on the exact angle between their ropes. The possible distances are approximately \( 1.3 \) meters, \( 7.9 \) meters, and \( 11.1 \) meters. 

Would you like more details on any specific part, or do you have any questions? Here are some related questions:

1. How would the distance change if Vishnu’s rope was 7 meters instead of 6.2 meters?
2. What is the impact on distance if the angle between the ropes is increased?
3. How does the Law of Cosines differ from the Law of Sines?
4. Can you provide a real-world scenario where this type of calculation is useful?
5. What would be the distance if both angles were 45 degrees?

**Tip:** When dealing with angles greater than 180 degrees, consider simplifying them to a range between 0 and 180 for easier calculations.

Learn how to calculate the distance between Vishnu and Athish using the Law of Cosines in a scenario involving ropes and angles. Explore multiple cases and understand the impact of different angles on the calculated distance. Suitable for high school students studying geometry and trigonometry.

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