To analyze the population growth of the local high school, you are given that the population in 2024 is 314 students and that it is growing at an annual rate of 1.18%.
Decay or Growth
Since the population is increasing, this represents growth.
r-value
The growth rate is typically expressed as a decimal in mathematical models. An annual growth rate of 1.18% can be converted to a decimal by dividing by 100:
\[ r = \frac{1.18}{100} = 0.0118 \]
Population Growth Formula
The population growth can be modeled using the exponential growth formula:
\[ P(t) = P_0 \cdot e^{rt} \]
Where:
- \( P(t) \) is the population at time \( t \)
- \( P_0 \) is the initial population (in this case, 314)
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
- \( r \) is the growth rate (0.0118)
- \( t \) is the time in years since the initial measurement.
However, if we want to use a simpler model that updates every year with a constant growth rate, we can use the following formula:
\[ P(t) = P_0 \cdot (1 + r)^t \]
Full Function Representation
Assuming the base year (the year we're modeling the growth from) is 2024 for \( t = 0 \), the function representing the population of the high school year by year can be expressed as:
\[ f(t) = 314 \cdot (1 + 0.0118)^t \]
Where \( t \) is the number of years after 2024 (e.g., \( t = 0 \) for 2024, \( t = 1 \) for 2025, etc.).
Summary
- The population is experiencing growth.
- The \( r \)-value is \( 0.0118 \).
- The function for the population over the years can be given by:
\[ f(t) = 314 \cdot (1.0118)^t \]
This function will allow you to compute the population for any year after 2024.