To determine which student is within the 12-kilometer radius of the network antenna, we need to calculate the distance from the antenna to each student using the Pythagorean theorem.

Information Given

• Carmen's Location: $2 \text{ km}$ west and $9 \text{ km}$ north.
• Manuel's Location: $4 \text{ km}$ east and $11 \text{ km}$ south.
• Network Radius: $12 \text{ km}$.

Steps to Solve

1. Calculate Carmen's Distance from the Antenna: $\text{Distance} = \sqrt{(2 \text{ km})^2 + (9 \text{ km})^2}$ $= \sqrt{4 + 81} = \sqrt{85} \approx 9.22 \text{ km}$

2. Calculate Manuel's Distance from the Antenna: $\text{Distance} = \sqrt{(4 \text{ km})^2 + (11 \text{ km})^2}$ $= \sqrt{16 + 121} = \sqrt{137} \approx 11.70 \text{ km}$

Conclusion

• Carmen is approximately $9.22 \text{ km}$ from the antenna, which is within the 12-kilometer range.
• Manuel is approximately $11.70 \text{ km}$ from the antenna, which is also within the 12-kilometer range.

Thus, both Carmen and Manuel are within the range of the antenna.

Would you like further details on any part of this solution, or have any questions?

Here are five related questions that expand upon this solution:

1. How would the calculation change if the antenna’s range were reduced to 10 kilometers?
2. How can we determine the coordinates of each student relative to a standard coordinate plane?
3. What is the formula for calculating the distance between two points in a plane?
4. How would the distances differ if they were measured in miles instead of kilometers?
5. How does the Pythagorean theorem help in calculating distances in a coordinate system?

Tip: Always double-check each calculation step when working with square roots and approximate values to ensure accuracy.