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 36 degrees 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 determine the distance between Vishnu and Athish, given the lengths of their ropes and the angles they make with the ground. This forms a triangle, where we can use the Law of Cosines to find the distance between them.

### Given:
- Vishnu’s rope length: \( 6.2 \) meters
- Angle between Vishnu’s rope and the ground: \( 36^\circ \)
- Athish’s rope length: \( 4.9 \) meters

### Step 1: Assumptions about the Angle Between Ropes
To calculate the distance between Vishnu and Athish, we need to know the angle between their ropes. We can analyze two main scenarios:

1. **Case 1:** Athish's rope is vertical, so the angle between Vishnu’s rope and Athish’s rope is \( 36^\circ \).
2. **Case 2:** Athish's rope makes an arbitrary angle with Vishnu’s rope. To find all possibilities, we'll explore a general angle \( \theta \) and a specific case where the angle is \( 180^\circ - 36^\circ = 144^\circ \).

### Step 2: Use the Law of Cosines
The distance \( d \) between Vishnu and Athish can be calculated using the Law of Cosines:

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

Where:
- \( a = 6.2 \) meters (Vishnu's rope)
- \( b = 4.9 \) meters (Athish's rope)
- \( \theta \) is the angle between the ropes.

### Case 1: \( \theta = 36^\circ \)
\[
d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(36^\circ)}
\]

\[
\cos(36^\circ) \approx 0.809
\]

\[
d = \sqrt{38.44 + 24.01 - 2 \times 6.2 \times 4.9 \times 0.809}
\]

\[
d = \sqrt{38.44 + 24.01 - 49.22} \approx \sqrt{13.23} \approx 3.6 \text{ meters}
\]

### Case 2: \( \theta = 144^\circ \)
\[
d = \sqrt{(6.2)^2 + (4.9)^2 - 2 \times 6.2 \times 4.9 \times \cos(144^\circ)}
\]

\[
\cos(144^\circ) \approx -0.809
\]

\[
d = \sqrt{38.44 + 24.01 - 2 \times 6.2 \times 4.9 \times (-0.809)}
\]

\[
d = \sqrt{38.44 + 24.01 + 49.22} \approx \sqrt{111.67} \approx 10.6 \text{ meters}
\]

### Summary of Possibilities:
- **When \( \theta = 36^\circ \):** The distance between Vishnu and Athish is approximately **3.6 meters**.
- **When \( \theta = 144^\circ \):** The distance between Vishnu and Athish is approximately **10.6 meters**.

### Diagrams:
1. **First Diagram**: The ropes form an acute angle (36 degrees) with the horizontal and nearly vertical lines, showing a smaller distance.
2. **Second Diagram**: The ropes form an obtuse angle (144 degrees), showing a larger distance.

### Conclusion:
Depending on the angle between the ropes, Vishnu and Athish could be either **3.6 meters** or **10.6 meters** apart. These calculations assume that Athish's rope is vertical for simplicity, or that we are examining the main possible angles between the ropes.

Would you like more details or have any questions? Here are some related questions:

1. How would the distance change if the angle between Vishnu and Athish's ropes was 90 degrees?
2. What other angles between the ropes could we consider?
3. How does changing the length of one rope affect the distance between Vishnu and Athish?
4. What would happen if the ropes were perpendicular to each other?
5. How would this problem change if the angle given was larger than 180 degrees?

**Tip:** When working with angles in triangles, the Law of Cosines is particularly useful for non-right triangles where we know two sides and the included angle.

Explore how to calculate the distance between Vishnu and Athish in a scientific study of thunderclouds. Given rope lengths and angles, use trigonometry and the Law of Cosines to find the distance accurately. Suitable for advanced high school students.

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