To solve the problem, we need to find an equation that models the population growth in the town over the given years using an exponential regression equation. The general form of an exponential function is:
[tex]\[ f(x) = a \cdot e^{b \cdot x} \][/tex]
We are given the following data points:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Years (x)} & \text{Population (f(x))} \\
\hline
0 & 10,500 \\
\hline
5 & 16,000 \\
\hline
10 & 26,000 \\
\hline
15 & 40,000 \\
\hline
20 & 65,000 \\
\hline
\end{array}
\][/tex]
Through the process of exponential regression, we can determine the constants [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that best fit this data set.
The result of our calculation gives us the values:
[tex]\[ a \approx 10134.7 \][/tex]
[tex]\[ b \approx 0.1 \][/tex]
Thus, the exponential equation that models the data is:
[tex]\[ f(x) = 10134.7 \cdot e^{0.1 \cdot x} \][/tex]
Next, we need to use this equation to estimate the population after 25 years. We substitute [tex]\( x = 25 \)[/tex] into our equation:
[tex]\[ f(25) = 10134.7 \cdot e^{0.1 \cdot 25} \][/tex]
Calculating this yields an estimated population of approximately 102,905 people.
To conclude:
1. The exponential regression equation that models the data is:
[tex]\[ f(x) = 10134.7 \cdot e^{0.1 \cdot x} \][/tex]
2. The estimated population after 25 years is:
[tex]\[ 102,905 \text{ people} \][/tex]
