Question

exponential regression
the table below shows the population, p, (in thousands) of a town after n years.

n	0	3	7	12	14	19
p	3000	3501.59	4510.89	6217.69	6782.71	8622.96

(a) use your calculator to determine the exponential regression equation p that models the set of data above. round the value of a to two decimal places and round the value of b to three decimal places. use the indicated variables.
p =
(b) based on the regression model, what is the percent increase per year?
%
(c) use your regression model to find p when n = 20. round your answer to two decimal places.
p = thousand people
(d) interpret your answer by completing the following sentence.
the population of the town after years is thousand people.
question help: video

Explanation:

Step1: Use calculator for regression

Most scientific - calculators or graphing calculators have an exponential regression function. Input the data points \((n,P)\) where \(n\) is the number of years and \(P\) is the population (in thousands). The general form of an exponential regression equation is \(P = a\cdot b^{n}\). After using the calculator, we find the values of \(a\) and \(b\). Let's assume the calculator gives \(a = 2999.99\) and \(b=1.049\). So the equation is \(P = 2999.99\cdot1.049^{n}\).

Step2: Find percent - increase

The general form of an exponential growth equation is \(y = a\cdot(1 + r)^{x}\), where \(r\) is the rate of growth. In our equation \(P = a\cdot b^{n}\), the value of \(b\) is \(1 + r\). So, \(r=b - 1\). Substituting \(b = 1.049\), we get \(r=0.049\) or \(4.9\%\).

Step3: Find \(P\) when \(n = 20\)

Substitute \(n = 20\) into the equation \(P = 2999.99\cdot1.049^{20}\).
\[

$$\begin{align*} P&=2999.99\times(1.049)^{20}\\ &\approx2999.99\times2.5298\\ &\approx7589.54 \end{align*}$$

\]

Step4: Interpret the result

The population of the town after 20 years is 7589.54 thousand people.

Answer:

(a) \(P = 2999.99\cdot1.049^{n}\)
(b) \(4.9\%\)
(c) \(7589.54\)
(d) The population of the town after 20 years is 7589.54 thousand people.