Welcome to Westonci.ca, where you can find answers to all your questions from a community of experienced professionals. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform. Explore comprehensive solutions to your questions from a wide range of professionals on our user-friendly platform.
Sagot :
To answer the given question, let's follow the steps one by one:
### 1. Find the regression equation for the rabbit population as a function of time \( x \).
We are given the data for the number of rabbits at different times in months:
- Time (months): [0, 3, 6, 9, 12]
- No. of Rabbits: [6, 32, 107, 309, 770]
We need to find the regression equation that models the population growth. Given the nature of the data, an exponential growth model will be suitable:
[tex]\[ P(t) = a \cdot e^{bt} \][/tex]
Where:
- \( P(t) \) is the population at time \( t \),
- \( a \) and \( b \) are constants to be determined.
Through regression analysis, we determine the values of \( a \) and \( b \). The optimal parameters for this model are found to be approximately:
[tex]\[ a \approx 16.70809436 \][/tex]
[tex]\[ b \approx 0.31952958 \][/tex]
### 2. Write the regression equation in terms of base \( e \).
Using the values of \( a \) and \( b \) derived from the regression analysis, the regression equation for the rabbit population as a function of time \( t \) is:
[tex]\[ P(t) = 16.70809436 \cdot e^{0.31952958t} \][/tex]
### 3. Use the equation from part (2) to estimate the time for the rabbits to exceed 10,000.
We need to find the time \( t \) when the rabbit population \( P(t) \) exceeds 10,000. So, we set up the equation:
[tex]\[ 10000 = 16.70809436 \cdot e^{0.31952958t} \][/tex]
First, isolate the exponential term:
[tex]\[ \frac{10000}{16.70809436} = e^{0.31952958t} \][/tex]
[tex]\[ 598.422 \approx e^{0.31952958t} \][/tex]
Next, take the natural logarithm of both sides to solve for \( t \):
[tex]\[ \ln(598.422) = 0.31952958t \][/tex]
[tex]\[ t = \frac{\ln(598.422)}{0.31952958} \][/tex]
Estimating the values:
[tex]\[ t \approx \frac{6.395}{0.31952958} \][/tex]
[tex]\[ t \approx 20.01206603670371 \][/tex]
Therefore, the time for the rabbit population to exceed 10,000 is approximately 20 months.
### 1. Find the regression equation for the rabbit population as a function of time \( x \).
We are given the data for the number of rabbits at different times in months:
- Time (months): [0, 3, 6, 9, 12]
- No. of Rabbits: [6, 32, 107, 309, 770]
We need to find the regression equation that models the population growth. Given the nature of the data, an exponential growth model will be suitable:
[tex]\[ P(t) = a \cdot e^{bt} \][/tex]
Where:
- \( P(t) \) is the population at time \( t \),
- \( a \) and \( b \) are constants to be determined.
Through regression analysis, we determine the values of \( a \) and \( b \). The optimal parameters for this model are found to be approximately:
[tex]\[ a \approx 16.70809436 \][/tex]
[tex]\[ b \approx 0.31952958 \][/tex]
### 2. Write the regression equation in terms of base \( e \).
Using the values of \( a \) and \( b \) derived from the regression analysis, the regression equation for the rabbit population as a function of time \( t \) is:
[tex]\[ P(t) = 16.70809436 \cdot e^{0.31952958t} \][/tex]
### 3. Use the equation from part (2) to estimate the time for the rabbits to exceed 10,000.
We need to find the time \( t \) when the rabbit population \( P(t) \) exceeds 10,000. So, we set up the equation:
[tex]\[ 10000 = 16.70809436 \cdot e^{0.31952958t} \][/tex]
First, isolate the exponential term:
[tex]\[ \frac{10000}{16.70809436} = e^{0.31952958t} \][/tex]
[tex]\[ 598.422 \approx e^{0.31952958t} \][/tex]
Next, take the natural logarithm of both sides to solve for \( t \):
[tex]\[ \ln(598.422) = 0.31952958t \][/tex]
[tex]\[ t = \frac{\ln(598.422)}{0.31952958} \][/tex]
Estimating the values:
[tex]\[ t \approx \frac{6.395}{0.31952958} \][/tex]
[tex]\[ t \approx 20.01206603670371 \][/tex]
Therefore, the time for the rabbit population to exceed 10,000 is approximately 20 months.
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.