Question

A small town has two local high schools. High School A currently has 900 students and is projected to grow by 50 students each year. High School B currently has 700 students and is projected to grow by 100 students each year. Let AA represent the number of students in High School A in tt years, and let BB represent the number of students in High School B after tt years. Graph each function and determine which high school is projected to have more students in 2 years.

Answer 1

To analyze the projected growth of both high schools over time, we can define the functions \( A(t) \) and \( B(t) \) based on the information given.

1. **High School A**:
- Current number of students: 900
- Growth rate: 50 students per year

The function can be defined as:
\[
A(t) = 900 + 50t
\]

2. **High School B**:
- Current number of students: 700
- Growth rate: 100 students per year

The function can be defined as:
\[
B(t) = 700 + 100t
\]

Next, we will calculate the number of students in each high school after \( t = 2 \) years.

### Calculate \( A(2) \) and \( B(2) \):
1. **For High School A**:
\[
A(2) = 900 + 50(2) = 900 + 100 = 1000
\]

2. **For High School B**:
\[
B(2) = 700 + 100(2) = 700 + 200 = 900
\]

### Conclusion:
After 2 years:
- High School A is projected to have 1000 students.
- High School B is projected to have 900 students.

Therefore, **High School A** is projected to have more students than **High School B** after 2 years.

### Graphing the Functions:
To visualize this, you could graph both functions \( A(t) \) and \( B(t) \) on a coordinate plane with \( t \) (years) on the x-axis and the number of students on the y-axis.

- **Function \( A(t) \)** would be a straight line starting from \( (0, 900) \) with a slope of 50.
- **Function \( B(t) \)** would also be a straight line starting from \( (0, 700) \) with a slope of 100.

The graph would show that \( B(t) \) rises faster than \( A(t) \) but started at a lower point. Over time, intersections or comparisons can be drawn at various points, including \( t = 2 \).

Overall, the growth trends suggest that although High School B has a higher growth rate per year, High School A’s larger starting population allows it to have more students after 2 years.