Answer :
To determine the initial population size and the population size after 9 years for the species, we use the provided population function:
[tex]\[ P(t) = \frac{340}{1 + 5e^{-0.31t}} \][/tex]
### Step-by-Step Solution
#### Initial Population Size
1. Determine the initial time value:
The initial population size corresponds to the time t = 0 years.
2. Substitute [tex]\( t = 0 \)[/tex] into the population function:
[tex]\[ P(0) = \frac{340}{1 + 5e^{-0.31 \cdot 0}} \][/tex]
3. Simplify the exponent term:
[tex]\[ e^{-0.31 \cdot 0} = e^0 = 1 \][/tex]
4. Calculate the population size at [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = \frac{340}{1 + 5 \cdot 1} = \frac{340}{1 + 5} = \frac{340}{6} \][/tex]
5. Divide and round to the nearest whole number:
[tex]\[ \frac{340}{6} \approx 56.67 \][/tex]
Rounding to the nearest whole number gives us:
[tex]\[ P(0) \approx 57 \][/tex]
Thus, the initial population size is 57 individuals.
#### Population Size After 9 Years
1. Determine the time value:
We need to find the population size at [tex]\( t = 9 \)[/tex] years.
2. Substitute [tex]\( t = 9 \)[/tex] into the population function:
[tex]\[ P(9) = \frac{340}{1 + 5e^{-0.31 \cdot 9}} \][/tex]
3. Calculate the exponent:
[tex]\[ -0.31 \cdot 9 = -2.79 \][/tex]
[tex]\[ e^{-2.79} \][/tex]
4. Approximate [tex]\( e^{-2.79} \)[/tex] using a calculator:
[tex]\[ e^{-2.79} \approx 0.0611 \][/tex]
5. Substitute this value back into the population function:
[tex]\[ P(9) = \frac{340}{1 + 5 \cdot 0.0611} = \frac{340}{1 + 0.3055} = \frac{340}{1.3055} \][/tex]
6. Divide and round to the nearest whole number:
[tex]\[ \frac{340}{1.3055} \approx 260.48 \][/tex]
Rounding to the nearest whole number gives us:
[tex]\[ P(9) \approx 260 \][/tex]
Thus, the population size after 9 years is 260 individuals.
### Summary
- Initial population size: 57 individuals
- Population size after 9 years: 260 individuals
[tex]\[ P(t) = \frac{340}{1 + 5e^{-0.31t}} \][/tex]
### Step-by-Step Solution
#### Initial Population Size
1. Determine the initial time value:
The initial population size corresponds to the time t = 0 years.
2. Substitute [tex]\( t = 0 \)[/tex] into the population function:
[tex]\[ P(0) = \frac{340}{1 + 5e^{-0.31 \cdot 0}} \][/tex]
3. Simplify the exponent term:
[tex]\[ e^{-0.31 \cdot 0} = e^0 = 1 \][/tex]
4. Calculate the population size at [tex]\( t = 0 \)[/tex]:
[tex]\[ P(0) = \frac{340}{1 + 5 \cdot 1} = \frac{340}{1 + 5} = \frac{340}{6} \][/tex]
5. Divide and round to the nearest whole number:
[tex]\[ \frac{340}{6} \approx 56.67 \][/tex]
Rounding to the nearest whole number gives us:
[tex]\[ P(0) \approx 57 \][/tex]
Thus, the initial population size is 57 individuals.
#### Population Size After 9 Years
1. Determine the time value:
We need to find the population size at [tex]\( t = 9 \)[/tex] years.
2. Substitute [tex]\( t = 9 \)[/tex] into the population function:
[tex]\[ P(9) = \frac{340}{1 + 5e^{-0.31 \cdot 9}} \][/tex]
3. Calculate the exponent:
[tex]\[ -0.31 \cdot 9 = -2.79 \][/tex]
[tex]\[ e^{-2.79} \][/tex]
4. Approximate [tex]\( e^{-2.79} \)[/tex] using a calculator:
[tex]\[ e^{-2.79} \approx 0.0611 \][/tex]
5. Substitute this value back into the population function:
[tex]\[ P(9) = \frac{340}{1 + 5 \cdot 0.0611} = \frac{340}{1 + 0.3055} = \frac{340}{1.3055} \][/tex]
6. Divide and round to the nearest whole number:
[tex]\[ \frac{340}{1.3055} \approx 260.48 \][/tex]
Rounding to the nearest whole number gives us:
[tex]\[ P(9) \approx 260 \][/tex]
Thus, the population size after 9 years is 260 individuals.
### Summary
- Initial population size: 57 individuals
- Population size after 9 years: 260 individuals
