Question

A new high school opens in a thriving suburban community. The number of students signed up to attend the school in its first year is 532, and the population of students is predicted to increase at a rate of 16% per year.

Which inequality can the high school use to determine the number of years, t, until the student population is larger than 2,025?

Answer 1

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.