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Answer :
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.
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.
Thanks for taking the time to read 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. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!
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