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The table below shows the population, [tex]\( P \)[/tex], (in thousands) of a town after [tex]\( n \)[/tex] years.
\begin{tabular}{|l|r|r|r|r|r|r|}
\hline
[tex]$n$[/tex] & 0 & 3 & 7 & 12 & 14 & 19 \\
\hline
[tex]$P$[/tex] & 4100 & 4922.22 & 6583.7 & 9511.01 & 10571.99 & 14086.37 \\
\hline
\end{tabular}

(a) Use your calculator to determine the exponential regression equation [tex]\( P \)[/tex] that models the set of data above. Round the value of [tex]\( a \)[/tex] to two decimal places and round the value of [tex]\( b \)[/tex] to three decimal places. Use the indicated variables.

[tex]\[ P = \square \][/tex]

(b) Based on the regression model, what is the percent increase per year?

[tex]\[ \square \% \][/tex]

(c) Use your regression model to find [tex]\( P \)[/tex] when [tex]\( n = 15 \)[/tex]. Round your answer to two decimal places.

[tex]\[ P = \square \][/tex] thousand people

(d) Interpret your answer by completing the following sentence.

The population of the town after [tex]\[ \square \][/tex] years is [tex]\[ \square \][/tex] thousand people.


Sagot :

Let's address each part of the problem step-by-step:

### Part (a)
We need to determine the exponential regression equation [tex]\(P = a \times b^n\)[/tex] that models the given set of data.

From the calculations, we have:
- [tex]\(a = 4115.03\)[/tex] (rounded to two decimal places)
- [tex]\(b = 1.069\)[/tex] (rounded to three decimal places)

Thus, the exponential regression equation is:
[tex]\[ P = 4115.03 \times 1.069^n \][/tex]

### Part (b)
To find the percent increase per year based on the regression model, we consider the base [tex]\(b\)[/tex] of the exponential equation:
- [tex]\(b = 1.069\)[/tex]

The percent increase is calculated as [tex]\((b - 1) \times 100 \%\)[/tex]:
[tex]\[ \text{Percent Increase} = (1.069 - 1) \times 100 \% = 0.069 \times 100 \% = 6.88 \% \][/tex]

### Part (c)
Next, we use the regression model to find the population [tex]\(P\)[/tex] when [tex]\(n = 15\)[/tex]:
[tex]\[ P = 4115.03 \times 1.069^{15} \][/tex]

From the calculations, when [tex]\(n = 15\)[/tex], we have [tex]\(P \approx 11157.37\)[/tex] thousand people (rounded to two decimal places).

### Part (d)
Finally, we interpret the result. We need to fill in the sentence correctly.

The sentence to complete is:
"The population of the town after 15 years is 11157.37 thousand people."

So, after [tex]\(n = 15\)[/tex] years, the population [tex]\(P\)[/tex] will be [tex]\(11157.37\)[/tex] thousand people.

### Summary
1. The exponential regression equation is:
[tex]\[ P = 4115.03 \times 1.069^n \][/tex]

2. The percent increase per year is:
[tex]\[ 6.88 \% \][/tex]

3. When [tex]\(n = 15\)[/tex], the population [tex]\(P\)[/tex] is:
[tex]\[ P = 11157.37 \][/tex] thousand people

4. Interpretation:
"The population of the town after 15 years is 11157.37 thousand people."

This completes the detailed step-by-step solution.
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