- Identify the initial population \(P_0 = 53,792\) and the growth rate \(r = 0.025\).
- Recognize that constant annual percentage growth follows the exponential function \(f(t) = P_0(1 + r)^t\).
- Substitute the values into the formula to obtain \(f(t) = 53,792(1.025)^t\).
- Select the matching function: \(\boxed{f(t) = 53,792(1.025)^t}\).
Explanation
- Analysis of the Growth Model
The problem asks us to model the population growth of Macon County after the year 2015. We are given the initial population in 2015, \(P_0 = 53,792\), and the constant annual growth rate, \(r = 2.5\%\). Let \(t\) represent the number of years after 2015. Since the population grows by a fixed percentage each year, this is a classic case of exponential growth. The standard form for an exponential growth function is \(f(t) = P_0(1 + r)^t\), where \(P_0\) is the initial amount, \(r\) is the growth rate expressed as a decimal, and \(t\) is the time elapsed.
- Formulating the Function
First, we convert the annual growth rate from a percentage to a decimal:
\[r = 2.5\% = \frac{2.5}{100} = 0.025.\]
Next, we determine the growth factor, which is the base of our exponent in the model:
\[1 + r = 1 + 0.025 = 1.025.\]
Now, we substitute the initial population \(P_0 = 53,792\) and the growth factor \(1.025\) into the general exponential growth formula \(f(t) = P_0(1 + r)^t\). This gives us the function:
\[f(t) = 53,792(1.025)^t.\]
- Conclusion and Selection
We compare our derived function \(f(t) = 53,792(1.025)^t\) with the given choices:
- \(f(t) = 53,792(1.25)^t\)
- \(f(t) = 53,792(1.025)^t\)
- \(f(t) = 1.25t + 53,792\)
- \(f(t) = 1.025t + 53,792\)
Our derived function matches the second option. Thus, the correct function to model the population \(t\) years after 2015 is \(f(t) = 53,792(1.025)^t\), and the correct choice is \(\boxed{f(t) = 53,792(1.025)^t}\).
Examples
Exponential growth models like this one are essential in real-world scenarios beyond population studies. For example, if you invest \(\$1,000\) in a savings account that earns a constant annual compound interest rate of \(3\%\), the balance of your account after \(t\) years can be modeled using the same structure: \(B(t) = 1000(1 + 0.03)^t = 1000(1.03)^t\). Understanding these models helps individuals make informed financial decisions and allows organizations to project future trends accurately based on current data.