Question

Avani and Nabhitha are both driving along the same highway in two different cars to a stadium in a distant city. At noon, Avani is 140 miles away from the stadium and Nabhitha is 200 miles away from the stadium. Avani is driving along the highway at a speed of 41 miles per hour and Nabhitha is driving at speed of 61 miles per hour. Let AA represent Avani's distance, in miles, away from the stadium tt hours after noon. Let NN represent Nabhitha's distance, in miles, away from the stadium tt hours after noon. Write an equation for each situation, in terms of t, commat, and determine whether Avani or Nabhitha is closer to the stadium 2 hours after noon.

Answer 1

To set up the equations for Avani's and Nabhitha's distances from the stadium after \( t \) hours, we start with their initial distances and their speeds.

### Avani's Situation:
- Initial distance from the stadium: 140 miles
- Speed: 41 miles per hour

After \( t \) hours, Avani's distance \( A(t) \) from the stadium can be expressed as:
\[
A(t) = 140 - 41t
\]
This represents that as time increases, Avani's distance from the stadium decreases due to her driving towards it.

### Nabhitha's Situation:
- Initial distance from the stadium: 200 miles
- Speed: 61 miles per hour

After \( t \) hours, Nabhitha's distance \( N(t) \) from the stadium can be expressed as:
\[
N(t) = 200 - 61t
\]
Similarly, this shows that as time passes, Nabhitha's distance from the stadium decreases as she is also driving towards it.

### Evaluation After 2 Hours:
To see who is closer to the stadium after 2 hours, we will calculate \( A(2) \) and \( N(2) \).

For Avani:
\[
A(2) = 140 - 41 \times 2 = 140 - 82 = 58 \text{ miles}
\]

For Nabhitha:
\[
N(2) = 200 - 61 \times 2 = 200 - 122 = 78 \text{ miles}
\]

### Conclusion:
After 2 hours, Avani is 58 miles away from the stadium, and Nabhitha is 78 miles away from the stadium.

**Therefore, Avani is closer to the stadium than Nabhitha 2 hours after noon.**