To find the initial population size and the population size after 7 years, we need to evaluate the function [tex]\( P(t) = \frac{1000}{1+7 e^{-0.42 t}} \)[/tex] at specific values of [tex]\( t \)[/tex].
### Initial Population Size
The initial population size corresponds to [tex]\( t = 0 \)[/tex].
1. Substitute [tex]\( t = 0 \)[/tex] into the function:
[tex]\[
P(0) = \frac{1000}{1+7 e^{-0.42 \cdot 0}}
\][/tex]
2. Simplify the exponent:
[tex]\[
P(0) = \frac{1000}{1+7 e^{0}}
\][/tex]
3. Knowing that [tex]\( e^0 = 1 \)[/tex]:
[tex]\[
P(0) = \frac{1000}{1+7 \cdot 1}
\][/tex]
4. Simplify further:
[tex]\[
P(0) = \frac{1000}{1+7} = \frac{1000}{8}
\][/tex]
5. Calculate the fraction:
[tex]\[
P(0) = \frac{1000}{8} = 125
\][/tex]
Therefore, the initial population size is 125 fish.
### Population Size After 7 Years
Next, we find the population size after 7 years, which corresponds to [tex]\( t = 7 \)[/tex].
1. Substitute [tex]\( t = 7 \)[/tex] into the function:
[tex]\[
P(7) = \frac{1000}{1+7 e^{-0.42 \cdot 7}}
\][/tex]
2. Multiply the exponent:
[tex]\[
P(7) = \frac{1000}{1+7 e^{-2.94}}
\][/tex]
3. Calculate [tex]\( e^{-2.94} \)[/tex] which is a small number since the exponent is negative.
4. Knowing the denominator involves adding a small [tex]\( 7 e^{-2.94} \)[/tex] to 1, we find:
[tex]\[
P(7) = \frac{1000}{1 + (value \approx 7e^{-2.94})}
\][/tex]
5. The resulting fraction when evaluated accurately gives a population size:
[tex]\[
P(7) \approx 730
\][/tex]
Therefore, the population size after 7 years is approximately 730 fish.
### Summary
- Initial population size: 125 fish
- Population size after 7 years: 730 fish
