Get expert advice and community support for all your questions on IDNLearn.com. Find the solutions you need quickly and accurately with help from our knowledgeable community.
Sagot :
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.
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. IDNLearn.com has the answers you need. Thank you for visiting, and we look forward to helping you again soon.
