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Answer :
Sure, let's work through the problem step by step.
1. Determine the Exponential Regression Equation:
To model the population of Lehi over time with an exponential function, we assume the form of the function is [tex]\( P(t) = a \cdot b^t \)[/tex].
Based on the regression model, we have:
- The "a" value is approximately [tex]\( 7849.76 \)[/tex].
- The "b" value is approximately [tex]\( 0.938 \)[/tex].
So, the equation for the model is:
[tex]\[ P(t) = 7849.76 \cdot (0.938)^t \][/tex]
2. Calculate the Percent Decrease Per Year:
The percent decrease per year is based on the value of [tex]\( b \)[/tex]:
- Since [tex]\( b \)[/tex] is approximately [tex]\( 0.938 \)[/tex], the decrease factor is [tex]\( 1 - 0.938 = 0.062 \)[/tex].
To express this as a percentage, multiply by 100:
[tex]\[ 0.062 \times 100 = 6.2\% \][/tex]
So, the population decreases by approximately 6.2% each year.
3. Find [tex]\( P(35) \)[/tex]:
To find the population 35 years after 1880:
- Substitute [tex]\( t = 35 \)[/tex] into the model:
[tex]\[ P(35) = 7849.76 \cdot (0.938)^{35} \][/tex]
The population [tex]\( P(35) \)[/tex] is approximately 911.
Interpretation:
"The population of Lehi 35 years after 1880 was about 911."
4. Determine When the Population Reaches 370:
We need to find the value of [tex]\( t \)[/tex] when [tex]\( P(t) = 370 \)[/tex]:
[tex]\[ 370 = 7849.76 \cdot (0.938)^t \][/tex]
To find [tex]\( t \)[/tex], solve the equation:
[tex]\[ t \approx 50 \][/tex]
Interpretation:
"In 50 years after 1880, the population of Lehi was about 370."
5. Determine When the Population is Halved:
The initial population is 8200. Half of this is 4100. We need to find [tex]\( t \)[/tex] when [tex]\( P(t) = 4100 \)[/tex]:
[tex]\[ 4100 = 7849.76 \cdot (0.938)^t \][/tex]
To find [tex]\( t \)[/tex], solve the equation:
[tex]\[ t \approx 11 \][/tex]
Interpretation:
"The population of Lehi halved 11 years after 1880."
These solutions provide a step-by-step explanation based on the results given. If you have further questions or need clarification on any step, feel free to ask!
1. Determine the Exponential Regression Equation:
To model the population of Lehi over time with an exponential function, we assume the form of the function is [tex]\( P(t) = a \cdot b^t \)[/tex].
Based on the regression model, we have:
- The "a" value is approximately [tex]\( 7849.76 \)[/tex].
- The "b" value is approximately [tex]\( 0.938 \)[/tex].
So, the equation for the model is:
[tex]\[ P(t) = 7849.76 \cdot (0.938)^t \][/tex]
2. Calculate the Percent Decrease Per Year:
The percent decrease per year is based on the value of [tex]\( b \)[/tex]:
- Since [tex]\( b \)[/tex] is approximately [tex]\( 0.938 \)[/tex], the decrease factor is [tex]\( 1 - 0.938 = 0.062 \)[/tex].
To express this as a percentage, multiply by 100:
[tex]\[ 0.062 \times 100 = 6.2\% \][/tex]
So, the population decreases by approximately 6.2% each year.
3. Find [tex]\( P(35) \)[/tex]:
To find the population 35 years after 1880:
- Substitute [tex]\( t = 35 \)[/tex] into the model:
[tex]\[ P(35) = 7849.76 \cdot (0.938)^{35} \][/tex]
The population [tex]\( P(35) \)[/tex] is approximately 911.
Interpretation:
"The population of Lehi 35 years after 1880 was about 911."
4. Determine When the Population Reaches 370:
We need to find the value of [tex]\( t \)[/tex] when [tex]\( P(t) = 370 \)[/tex]:
[tex]\[ 370 = 7849.76 \cdot (0.938)^t \][/tex]
To find [tex]\( t \)[/tex], solve the equation:
[tex]\[ t \approx 50 \][/tex]
Interpretation:
"In 50 years after 1880, the population of Lehi was about 370."
5. Determine When the Population is Halved:
The initial population is 8200. Half of this is 4100. We need to find [tex]\( t \)[/tex] when [tex]\( P(t) = 4100 \)[/tex]:
[tex]\[ 4100 = 7849.76 \cdot (0.938)^t \][/tex]
To find [tex]\( t \)[/tex], solve the equation:
[tex]\[ t \approx 11 \][/tex]
Interpretation:
"The population of Lehi halved 11 years after 1880."
These solutions provide a step-by-step explanation based on the results given. If you have further questions or need clarification on any step, feel free to ask!
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