Exponential Regression

The table below shows the population of a fictional California Gold Rush town named Lehi in the years after its peak population in 1880.

\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Year & 1880 & 1890 & 1900 & 1910 & 1920 & 1930 \\
\hline
Population & 9800 & 5081 & 4331 & 3542 & 1914 & 1081 \\
\hline
\end{tabular}
\]

For the purpose of this problem, let [tex]$P$[/tex] represent the population of Lehi [tex]$t$[/tex] years after 1880 ([tex]$t=0$[/tex] represents 1880). The new table is:

\[
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
[tex]$t$[/tex] & 0 & 10 & 20 & 30 & 40 & 50 \\
\hline
[tex]$P(t)$[/tex] & 9800 & 5081 & 4331 & 3542 & 1914 & 1081 \\
\hline
\end{tabular}
\]

1. Use your calculator to determine the exponential regression equation that models the set of data above. Round the [tex]$a$[/tex] value to two decimals, and round the [tex]$b$[/tex] value to two decimals. Use the indicated variables and proper function notation.

[tex]\[ P(t) = \][/tex] [tex]\[\square\][/tex]

2. Based on your regression model, what is the percent decrease per year?

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

3. Find [tex]$P(35)$[/tex]. Round your answer to the nearest whole number.

[tex]\[ P(35) = \][/tex] [tex]\[\square\][/tex]

4. Interpret your answer by completing the following sentence. Be sure to use units in your answer.

"The population of Lehi [tex]$\square$[/tex] after 1880 was about [tex]$\square$[/tex]."

5. How long did it take for the population of Lehi to reach 350 people? Round your answer to the nearest whole number.

[tex]\[ P(t) = 350 \text{ when } t = \][/tex] [tex]\[\square\][/tex]

To determine the exponential regression equation that best models the given data, we follow these steps:

1. Data Preparation:
Given data for the years after 1880:

| [tex]$t$[/tex] (years after 1880) | 0 | 10 | 20 | 30 | 40 | 50 |
|---------------------------|---|----|----|----|----|----|
| [tex]$P(t)$[/tex] (population) | 9800 | 5081 | 4331 | 3542 | 1914 | 1081 |

2. Exponential Model:
The form of the exponential regression model we are fitting is
[tex]\[
P(t) = a \cdot e^{bt}
\][/tex]

3. Parameters:
After computing the best fit using exponential regression, we obtain:
[tex]\[
a \approx 0.00 \quad \text{and} \quad b \approx 1.00
\][/tex]
Thus, the exponential regression model is:
[tex]\[
P(t) = 0.00 \cdot e^{1.00t}
\][/tex]

4. Percent Decrease Per Year:
To find the percent decrease per year, we use the formula:
[tex]\[
\text{Percent Decrease Per Year} = (1 - e^b) \times (-100) \%
\][/tex]
Plugging in the value of [tex]$ b $[/tex]:
[tex]\[
\text{Percent Decrease Per Year} \approx 171.83 \%
\][/tex]

5. Population at [tex]$ t = 35 $[/tex]:
Using the regression model to find the population at [tex]$ t = 35 $[/tex]:
[tex]\[
P(35) = 0.00 \cdot e^{1.00 \times 35} = 0
\][/tex]
Rounding to the nearest whole number:
[tex]\[
P(35) \approx 0
\][/tex]

6. Interpretation of [tex]$ P(35) $[/tex]:
"The population of Lehi 35 years after 1880 was about 0."

7. Time to Reach Population of 350:
To find the time [tex]$ t $[/tex] when the population [tex]$ P(t) = 350 $[/tex]:
[tex]\[
350 = a \cdot e^{bt}
\][/tex]
Solving for [tex]$ t $[/tex]:
[tex]\[
t = \frac{\ln(350/a)}{b}
\][/tex]
Using [tex]$ a \approx 0.00 $[/tex] and [tex]$ b \approx 1.00 $[/tex]:
[tex]\[
t \approx 49
\][/tex]
So it takes approximately 49 years for the population to reach 350 people.

In summary:

1. The exponential regression model is:
[tex]\[
P(t) = 0.00 \cdot e^{1.00t}
\][/tex]

2. The percent decrease per year is approximately:
[tex]\[
171.83 \%
\][/tex]

3. The population at [tex]$ t = 35 $[/tex] years is:
[tex]\[
P(35) = 0
\][/tex]

4. It takes approximately:
[tex]\[
P(t) = 350 \text{ when } t = 49
\][/tex] years for the population to reach 350 people.