Whether you're a student or a professional, IDNLearn.com has answers for everyone. Our platform provides trustworthy answers to help you make informed decisions quickly and easily.
Sagot :
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.
### 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.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for choosing IDNLearn.com for your queries. We’re committed to providing accurate answers, so visit us again soon.