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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.
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