Questions: In 2020, there were 13,800 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 20,800 students at college B, with a projected enrollment decline of 1250 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time?

In 2020, there were 13,800 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 20,800 students at college B, with a projected enrollment decline of 1250 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time?

Transcript text: In 2020, there were 13,800 students at college A, with a projected enrollment increase of 500 students per year. In the same year, there were 20,800 students at college B, with a projected enrollment decline of 1250 students per year. According to these projections, when will the colleges have the same enrollment? What will be the enrollment in each college at that time?

Solution

Solution Steps

To determine when the enrollments at both colleges will be the same, we need to set up equations for the enrollment of each college as a function of time and solve for the year when these two equations are equal.

Define the enrollment functions for both colleges.
Set the two functions equal to each other to find the year when the enrollments are the same.
Solve the resulting equation for the year.
Calculate the enrollment at that time.

Step 1: Define Enrollment Functions

Let \( E_A(t) \) and \( E_B(t) \) represent the enrollments at colleges A and B, respectively, as functions of time \( t \) (in years after 2020).

\[ E_A(t) = 13800 + 500t \] \[ E_B(t) = 20800 - 1250t \]

Step 2: Set the Enrollments Equal

To find the year when the enrollments are the same, set \( E_A(t) \) equal to \( E_B(t) \):

\[ 13800 + 500t = 20800 - 1250t \]

Step 3: Solve for \( t \)

Rearrange the equation to solve for \( t \):

\[ 13800 + 500t = 20800 - 1250t \] \[ 500t + 1250t = 20800 - 13800 \] \[ 1750t = 7000 \] \[ t = \frac{7000}{1750} = 4 \]

Step 4: Calculate the Enrollment at That Time

Substitute \( t = 4 \) back into either enrollment function to find the enrollment at that time:

\[ E_A(4) = 13800 + 500 \times 4 = 13800 + 2000 = 15800 \]