Question

question 19
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	2600	3034.71	3909.44	5388.66	5878.35	7473.23

(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.
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Explanation:

Step1: Use calculator for regression

Most scientific - 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 \(P = ab^{n}\). After using the calculator, we get \(a\approx2599.99\) and \(b\approx1.050\). So, \(P = 2599.99\times1.050^{n}\).

Step2: Find percent - increase

The formula for percent - increase in an exponential growth model \(y = ab^{x}\) is \((b - 1)\times100\%\). Here, \(b = 1.050\), so the percent - increase is \((1.050-1)\times100\%=5.00\%\).

Step3: Calculate \(P\) for \(n = 20\)

Substitute \(n = 20\) into the regression model \(P = 2599.99\times1.050^{20}\). First, calculate \(1.050^{20}\approx2.653297705\). Then, \(P=2599.99\times2.653297705\approx6909.74\).

Step4: Interpret the result

The population of the town after 20 years is 6909.74 thousand people.

Answer:

(a) \(P = 2599.99\times1.050^{n}\)
(b) \(5.00\)
(c) \(6909.74\)
(d) 20; 6909.74