Question

the table below shows the population of a town over x years.

years	population
5	16,000
10	26,000
15	40,000
20	65,000

what values, rounded to the nearest tenth, complete the exponential regression equation that models the data?
$f(x) = \square (\square)^x$
based on the regression equation and rounded to the nearest whole person, what is the estimated population after 25 years?
\square people

Explanation:

Step1: Use exponential regression formula

The general form of an exponential function is \(y = ab^{x}\). Using a calculator or software (like Excel's EXPON.DIST function or a graphing calculator's regression feature) to perform exponential regression on the data points \((x = 0,y = 10500)\), \((x = 5,y = 16000)\), \((x = 10,y = 26000)\), \((x = 15,y = 40000)\), \((x = 20,y = 65000)\).
When \(x = 0\), \(y=ab^{0}=a\), so \(a\approx10500.0\).
For \(b\), using the regression result, \(b\approx1.1\).

Step2: Find the population after \(x = 25\) years

Substitute \(a = 10500\), \(b = 1.1\) and \(x = 25\) into the formula \(y=ab^{x}\).
\(y=10500\times(1.1)^{25}\)
First, calculate \((1.1)^{25}\approx10.83470594\)
Then, \(y = 10500\times10.83470594\approx113764.4124\)

Answer:

\(f(x)=10500.0(1.1)^{x}\); \(113764\) people