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.
