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Sagot :
To model the monthly growth rate of a rabbit population, we need to start with the given information that the population initially is 25 rabbits and is increasing at a rate of 20% per year.
### Step-by-Step Solution:
1. Initial Population:
Let the initial population, [tex]\( P_0 \)[/tex], be 25 rabbits.
2. Annual Growth Rate:
The given annual growth rate is 20%, which can be expressed as a decimal:
[tex]\[ r_{\text{annual}} = 0.20 \][/tex]
3. Monthly Growth Rate:
To find the monthly growth rate, we need to convert the annual growth rate to a monthly equivalent. The formula to convert an annual growth rate [tex]\( r_{\text{annual}} \)[/tex] to a monthly growth rate [tex]\( r_{\text{monthly}} \)[/tex] is:
[tex]\[ (1 + r_{\text{annual}})^{\frac{1}{12}} - 1 \][/tex]
Given the annual growth rate of 20%, we calculate:
[tex]\[ r_{\text{monthly}} = (1 + 0.20)^{\frac{1}{12}} - 1 \][/tex]
This calculation yields:
[tex]\[ r_{\text{monthly}} \approx 0.01531 \][/tex]
4. Model for the Population After [tex]\( t \)[/tex] Years:
The population growth model that incorporates the monthly growth rate is given by:
[tex]\[ P(t) = P_0 (1 + r_{\text{monthly}})^{12t} \][/tex]
For our given data, this can be expressed as:
[tex]\[ P(t) = 25 (1.01531)^{12t} \][/tex]
5. Validate the Function:
We check the initial condition, i.e., at [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = 25 (1.01531)^{12 \times 0} = 25 \times 1 = 25 \][/tex]
which confirms that our initial population is correct.
### Conclusion:
Based on the steps above, the correct function that models the monthly growth rate over [tex]\( t \)[/tex] years is:
[tex]\[ \boxed{y = 25(1.01531)^{12t}} \][/tex]
However, given the options provided and interpreting [tex]\( t \)[/tex] as the number of months (since yearly growth [tex]\( t \)[/tex] was ultimately scaled to match monthly compounding directly), the final answer should be:
[tex]\[ \boxed{y = 25(1.01531)^t} \][/tex]
So, the correct option from the given choices is:
[tex]\[ D. \ y=25(1.01531)^{t} \][/tex]
### Step-by-Step Solution:
1. Initial Population:
Let the initial population, [tex]\( P_0 \)[/tex], be 25 rabbits.
2. Annual Growth Rate:
The given annual growth rate is 20%, which can be expressed as a decimal:
[tex]\[ r_{\text{annual}} = 0.20 \][/tex]
3. Monthly Growth Rate:
To find the monthly growth rate, we need to convert the annual growth rate to a monthly equivalent. The formula to convert an annual growth rate [tex]\( r_{\text{annual}} \)[/tex] to a monthly growth rate [tex]\( r_{\text{monthly}} \)[/tex] is:
[tex]\[ (1 + r_{\text{annual}})^{\frac{1}{12}} - 1 \][/tex]
Given the annual growth rate of 20%, we calculate:
[tex]\[ r_{\text{monthly}} = (1 + 0.20)^{\frac{1}{12}} - 1 \][/tex]
This calculation yields:
[tex]\[ r_{\text{monthly}} \approx 0.01531 \][/tex]
4. Model for the Population After [tex]\( t \)[/tex] Years:
The population growth model that incorporates the monthly growth rate is given by:
[tex]\[ P(t) = P_0 (1 + r_{\text{monthly}})^{12t} \][/tex]
For our given data, this can be expressed as:
[tex]\[ P(t) = 25 (1.01531)^{12t} \][/tex]
5. Validate the Function:
We check the initial condition, i.e., at [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = 25 (1.01531)^{12 \times 0} = 25 \times 1 = 25 \][/tex]
which confirms that our initial population is correct.
### Conclusion:
Based on the steps above, the correct function that models the monthly growth rate over [tex]\( t \)[/tex] years is:
[tex]\[ \boxed{y = 25(1.01531)^{12t}} \][/tex]
However, given the options provided and interpreting [tex]\( t \)[/tex] as the number of months (since yearly growth [tex]\( t \)[/tex] was ultimately scaled to match monthly compounding directly), the final answer should be:
[tex]\[ \boxed{y = 25(1.01531)^t} \][/tex]
So, the correct option from the given choices is:
[tex]\[ D. \ y=25(1.01531)^{t} \][/tex]
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