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. when will they intercept?

Answer 1

To solve the problem, we can define the equations for the students in both high schools over time \( t \), where \( t \) is measured in years.

### High School A:
- Current number of students: 900
- Growth rate: 50 students per year

So the equation for the number of students in High School A after \( t \) years, \( A(t) \), can be written as:
\[
A(t) = 900 + 50t
\]

### High School B:
- Current number of students: 700
- Growth rate: 100 students per year

So the equation for the number of students in High School B after \( t \) years, \( B(t) \), is:
\[
B(t) = 700 + 100t
\]

### Determine the Number of Students After 2 Years
Now we will calculate the number of students in each school after 2 years.

**For High School A:**
\[
A(2) = 900 + 50 \times 2 = 900 + 100 = 1000
\]

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

**Conclusion After 2 Years:**
- High School A will have 1000 students.
- High School B will have 900 students.

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

### Finding When the Two Schools Intercept
To find when the two schools will have the same number of students, we set \( A(t) \) equal to \( B(t) \):

\[
900 + 50t = 700 + 100t
\]

Now, we can solve for \( t \):

1. Subtract \( 50t \) from both sides:
\[
900 = 700 + 50t
\]

2. Subtract 700 from both sides:
\[
200 = 50t
\]

3. Divide both sides by 50:
\[
t = \frac{200}{50} = 4
\]

### Conclusion
The two high schools will intercept in 4 years.

### Summary:
- After 2 years, High School A will have 1000 students, and High School B will have 900 students.
- High School A is larger at that time.
- The two schools will have the same number of students in 4 years.