Let's carefully address each part of the question using the data provided, starting with the calculations to find the regression equation.
### 1. Regression Equation in Terms of a Base Other Than [tex]\( e \)[/tex]
Given the data:
| Time (months) | 0 | 3 | 6 | 9 | 12 |
|---------------|---|----|-----|-----|-----|
| No. of Rabbits| 6 | 32 | 107 | 309 | 770 |
We need to determine an exponential growth model of the form [tex]\( y = a \cdot b^x \)[/tex].
Upon performing the necessary regression analysis:
- We find that [tex]\( a \approx 7.898 \)[/tex].
- We also find that the growth rate [tex]\( b \approx 1.491 \)[/tex].
Hence, the regression equation is:
[tex]\[ y = 7.898 \times (1.491)^x \][/tex]
### 2. Regression Equation in Terms of Base [tex]\( e \)[/tex]
In exponential growth models, converting the base from some value [tex]\( b \)[/tex] to the natural base [tex]\( e \)[/tex] involves expressing the equation [tex]\( y = a \cdot b^x \)[/tex] in the form [tex]\( y = a \cdot e^{kx} \)[/tex].
Interestingly, we find:
- [tex]\( a \approx 7.898 \)[/tex]
- [tex]\( k \approx 0.3992 \)[/tex]
Therefore, the regression equation in terms of base [tex]\( e \)[/tex] is:
[tex]\[ y = 7.898 \times e^{0.3992x} \][/tex]
### 3. Estimating the Time for the Population to Exceed 10,000 Rabbits
From the model [tex]\( y = 7.898 \times e^{0.3992x} \)[/tex], we can solve for [tex]\( x \)[/tex] when [tex]\( y = 10,000 \)[/tex].
We need to solve for [tex]\( x \)[/tex] in:
[tex]\[ 10000 = 7.898 \times e^{0.3992x} \][/tex]
Rearranging the equation:
[tex]\[ \frac{10000}{7.898} = e^{0.3992x} \][/tex]
Taking the natural logarithm of both sides:
[tex]\[ \ln\left(\frac{10000}{7.898}\right) = 0.3992x \][/tex]
Thus:
[tex]\[ x = \frac{\ln(10000 / 7.898)}{0.3992} \][/tex]
Upon solving this, we determine that:
[tex]\[ x \approx 17.9 \text{ months} \][/tex]
### Summary
- The regression equation in terms of a base other than [tex]\( e \)[/tex] is:
[tex]\[ y = 7.898 \times (1.491)^x \][/tex]
- The regression equation in terms of base [tex]\( e \)[/tex] is:
[tex]\[ y = 7.898 \times e^{0.3992x} \][/tex]
- The estimated time for the rabbit population to exceed 10,000 is:
[tex]\[ x \approx 17.9 \text{ months} \][/tex]
These results correspond to the first set of answers provided in your question.
