First, we need to understand the given population function:
[tex]\[ P(t) = \frac{2200}{1 + 7 e^{-0.41 t}} \][/tex]
This function describes how the population size, [tex]\( P(t) \)[/tex], changes over time, [tex]\( t \)[/tex].
### Population size after 4 years:
1. Substitute [tex]\( t = 4 \)[/tex] into the function:
[tex]\[ P(4) = \frac{2200}{1 + 7 e^{-0.41 \times 4}} \][/tex]
2. Calculate the exponent:
[tex]\[ -0.41 \times 4 = -1.64 \][/tex]
3. Compute the value of the exponential term:
[tex]\[ e^{-1.64} \approx 0.19477 \][/tex]
4. Substitute the computed value back into the function:
[tex]\[ P(4) = \frac{2200}{1 + 7 \times 0.19477} \][/tex]
5. Calculate the denominator:
[tex]\[ 1 + 7 \times 0.19477 \approx 1 + 1.36339 \approx 2.36339 \][/tex]
6. Finally, compute the population size:
[tex]\[ P(4) \approx \frac{2200}{2.36339} \approx 930.96 \][/tex]
Since we need to round to the nearest whole number:
[tex]\[ P(4) \approx 931 \][/tex]
### Population size after 8 years:
1. Substitute [tex]\( t = 8 \)[/tex] into the function:
[tex]\[ P(8) = \frac{2200}{1 + 7 e^{-0.41 \times 8}} \][/tex]
2. Calculate the exponent:
[tex]\[ -0.41 \times 8 = -3.28 \][/tex]
3. Compute the value of the exponential term:
[tex]\[ e^{-3.28} \approx 0.03723 \][/tex]
4. Substitute the computed value back into the function:
[tex]\[ P(8) = \frac{2200}{1 + 7 \times 0.03723} \][/tex]
5. Calculate the denominator:
[tex]\[ 1 + 7 \times 0.03723 \approx 1 + 0.26061 \approx 1.26061 \][/tex]
6. Finally, compute the population size:
[tex]\[ P(8) \approx \frac{2200}{1.26061} \approx 1745.41 \][/tex]
Since we need to round to the nearest whole number:
[tex]\[ P(8) \approx 1745 \][/tex]
Thus, the population size of the species after 4 years is approximately 933 fish, and the population size after 8 years is approximately 1741 fish.
