To determine the number of years \( t \) until the student population is larger than 2,025, we can model the population growth using an exponential function.
The initial number of students (\( P_0 \)) is 532, and the population is expected to grow at a rate of 16% per year. The formula for the population after \( t \) years can be expressed as:
\[ P(t) = P_0(1 + r)^t \]
Where:
- \( P(t) \) is the population after \( t \) years,
- \( P_0 \) is the initial population (532),
- \( r \) is the growth rate (16% or 0.16),
- \( t \) is the number of years.
Plugging in the values, we get:
\[ P(t) = 532(1 + 0.16)^t \] \[ P(t) = 532(1.16)^t \]
We want to find the number of years \( t \) such that \( P(t) > 2025 \). Therefore, we set up the following inequality:
\[ 532(1.16)^t > 2025 \]
This is the inequality the high school can use to determine the number of years until the student population exceeds 2,025.
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