Certainly! Let's walk through this step-by-step:
### Part (a): The Model for the Population
The rabbit population increases by [tex]\(12 \%\)[/tex] per year. This means that each year, the population is multiplied by [tex]\(1.12\)[/tex]. If the initial population is 484, the population [tex]\(y\)[/tex] after [tex]\(t\)[/tex] years can be modeled by the following exponential growth function:
[tex]\[ y = 484 \times (1.12)^t \][/tex]
### Part (b): Finding the Rabbit Population in 13 Years
To find the rabbit population after 13 years, substitute [tex]\(t = 13\)[/tex] into the model:
[tex]\[ y = 484 \times (1.12)^{13} \][/tex]
By calculating this, we find that the rabbit population in 13 years is:
[tex]\[ y \approx 2112 \][/tex]
So, the rabbit population in 13 years is approximately 2112 rabbits.
### Part (c): Estimating When the Rabbit Population Reaches 17,215
To estimate the year when the rabbit population reaches 17,215, we need to solve the equation:
[tex]\[ 17215 = 484 \times (1.12)^t \][/tex]
We can rearrange this equation to isolate [tex]\(t\)[/tex]:
[tex]\[ (1.12)^t = \frac{17215}{484} \][/tex]
Now, we take the logarithm of both sides. Using the properties of logarithms, we get:
[tex]\[ t \approx \frac{\log(\frac{17215}{484})}{\log(1.12)} \][/tex]
By calculating this, we find that [tex]\(t\)[/tex] is approximately between 31 and 32.
So, the rabbit population will reach 17,215 between year 31 and year 32.
Thus, summarizing the solutions:
- The population model is [tex]\( y = 484 \times (1.12)^t \)[/tex].
- The rabbit population in 13 years is approximately 2112 rabbits.
- The population reaches 17,215 between year 31 and year 32.
