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?

1 answer

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.