A new applied mathematics program at a university has grown in enrollment by 14 students per year for the last five years. If the program enrolled 50 students in its third year, wh... A new applied mathematics program at a university has grown in enrollment by 14 students per year for the last five years. If the program enrolled 50 students in its third year, what is an equation for the number of students S in the program as a function of the number of years T since its inception?
Understand the Problem
The question is asking for the equation relating to students' enrollment in a mathematics program at a university over several years. It specifically inquires about finding the equation for the number of students as a function of the number of years since the program started, given specific rates of growth and initial enrollment.
Answer
The equation for enrollment is $S(T) = 14T + 36$.
Answer for screen readers
The equation for the number of students as a function of the number of years since its inception is $$ S(T) = 14T + 36 $$
Steps to Solve
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Identify the known values The problem states that the math program enrolled 50 students in its third year and has been growing by 14 students per year.
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Establish the year-to-time relationship Since we are looking for the function of the number of students $S$ as a function of the number of years $T$, we need to determine $T$ when the enrollment was 50 students:
- In Year 1: $S = 50 - 14 \cdot 1 = 36$
- In Year 2: $S = 50 - 14 \cdot 1 = 50$
- In Year 3: $S = 50$ (known directly)
- Formulate the linear equation The growth model can be formulated as: $$ S(T) = S_0 + r \cdot T $$ where:
- $S_0 = 36$ (the calculated enrollment in the first year)
- $r = 14$ (the rate of growth)
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Plug in the known values to create the equation Using the established values: $$ S(T) = 36 + 14 \cdot T $$
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State the function Finally, we explicitly state the equation relating to students' enrollment: $$ S(T) = 14T + 36 $$
The equation for the number of students as a function of the number of years since its inception is $$ S(T) = 14T + 36 $$
More Information
This equation indicates that the program started with 36 students and grows by 14 each year. Thus, after 5 years, the total enrollment would be $S(5) = 14 \cdot 5 + 36 = 106$ students.
Tips
- Forgetting the initial enrollment: It's important to accurately calculate the initial number of students before applying the growth rate.
- Misapplying the growth rate: Ensure that the growth rate is added each year correctly.
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