the table below shows the population of a town over x years.\n| years | population |\n| ---- | ---- |\n| 0 |…

Explanation:

Step1: Recall exponential - function form

The general form of an exponential function is $f(x)=ab^{x}$, where $a$ is the initial value and $b$ is the growth factor. When $x = 0$, $f(0)=a$. From the table, when $x = 0$, the population $f(0)=10500$, so $a = 10500$.

Step2: Find the value of $b$

We use another data - point, say $(x = 5,f(5)=16000)$. Substitute $a = 10500$, $x = 5$, and $f(5)=16000$ into $f(x)=ab^{x}$. We get $16000=10500b^{5}$. Then $b^{5}=\frac{16000}{10500}=\frac{32}{21}$. So $b=\left(\frac{32}{21}\right)^{\frac{1}{5}}\approx1.09$.

Step3: Write the exponential regression equation

The exponential regression equation is $f(x)=10500(1.09)^{x}$.

Step4: Estimate the population after 25 years

Substitute $x = 25$ into $f(x)=10500(1.09)^{x}$. We have $f(25)=10500\times(1.09)^{25}$. Using a calculator, $(1.09)^{25}\approx8.62308066$. Then $f(25)=10500\times8.62308066\approx90542$.

Answer:

$f(x)=10500(1.09)^{x}$; 90542