Question

the table below shows the population of a town over x years. what values, rounded to the nearest tenth, complete the exponential regression equation that models the data? based on the regression equation and rounded to the nearest whole person, what is the estimated population after 25 years? people

Explanation:

Step1: Use a calculator for exponential regression

Input the data points \((x,y)\) where \(x = 0,5,10,15,20\) and \(y=10500,16000,26000,40000,65000\) into a calculator with an exponential regression function.
The general form of an exponential function is \(y = ab^{x}\).
After performing the regression, we get \(a\approx10511.4\) and \(b\approx1.1\).

Step2: Find the population after 25 years

Substitute \(x = 25\), \(a = 10511.4\), and \(b=1.1\) into the equation \(y=ab^{x}\).

$$y=10511.4\times(1.1)^{25}$$

First, calculate \((1.1)^{25}\approx10.83470594\)
Then, \(y = 10511.4\times10.83470594\approx113999\)

Answer:

\(f(x)=10511.4(1.1)^{x}\), \(114000\) people