A species of fish was added to a lake. The population size [tex]$P(t)$[/tex] of this species can be modeled by the following function, where [tex]$t$[/tex] is the number of years from the time the species was added to the lake.

[tex]
P(t)=\frac{1000}{1+7 e^{-0.42 t}}
[/tex]

Find the initial population size of the species and the population size after 7 years. Round your answers to the nearest whole number as necessary.

- Initial population size: [tex]$\square$[/tex] fish
- Population size after 7 years: [tex]$\square$[/tex] fish



Answer :

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