determining the regression equation and making predictions\nthe table below shows the population of a town…

determining the regression equation and making predictions\nthe table below shows the population of a town over x years.\n| years | population |\n| ---- | ---- |\n| 0 | 10,500 |\n| 5 | 16,000 |\n| 10 | 26,000 |\n| 15 | 40,000 |\n| 20 | 65,000 |\nwhat values, rounded to the nearest tenth, complete the exponential regression equation that models the data?\nf(x) = ( )^x\nbased on the regression equation and rounded to the nearest whole person, what is the estimated population after 25 years?\n people

determining the regression equation and making predictions\nthe table below shows the population of a town over x years.\n| years | population |\n| ---- | ---- |\n| 0 | 10,500 |\n| 5 | 16,000 |\n| 10 | 26,000 |\n| 15 | 40,000 |\n| 20 | 65,000 |\nwhat values, rounded to the nearest tenth, complete the exponential regression equation that models the data?\nf(x) = ( )^x\nbased on the regression equation and rounded to the nearest whole person, what is the estimated population after 25 years?\n people

Answer

Explanation:

Step1: Recall exponential regression form

The general form of an exponential regression equation is $f(x)=ab^{x}$. 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 regression equation

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

Step4: Make a prediction

To find the population after 25 years, substitute $x = 25$ into $f(x)=10500(1.09)^{x}$. So $f(25)=10500\times(1.09)^{25}$. Using a calculator, $(1.09)^{25}\approx8.62308066$. Then $f(25)=10500\times8.62308066\approx90542$.

Answer:

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