A species of fish was added to a lake. The population size of this species can be modeled by the following function, where is the number of years from the
time the species was added to the lake.

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:
Population size after 7 years:

Ask by Morgan Murray. in the United States

Mar 28,2025

Question

A species of fish was added to a lake. The population size of this species can be modeled by the following function, where is the number of years from the
time the species was added to the lake.

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:
Population size after 7 years:

Ask by Morgan Murray. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

We are given the function

$$
P(t) = \frac{800}{1+5 e^{-0.4t}},
$$

where $$ t $$ is the number of years after the species was added.

### Step 1. Find the initial population ($$ t = 0 $$)
Substitute $$ t = 0 $$ into the function:

$$
P(0) = \frac{800}{1+5 e^{-0.4(0)}} = \frac{800}{1+5 e^{0}}.
$$

Since $$ e^0 = 1 $$, this simplifies to:

$$
P(0) = \frac{800}{1+5} = \frac{800}{6} \approx 133.33.
$$

Rounding to the nearest whole number:

$$
\text{Initial population size} \approx 133.
$$

### Step 2. Find the population size after 7 years ($$ t = 7 $$)
Substitute $$ t = 7 $$ into the function:

$$
P(7) = \frac{800}{1+5 e^{-0.4(7)}} = \frac{800}{1+5 e^{-2.8}}.
$$

Next, approximate $$ e^{-2.8} $$:

$$
e^{-2.8} \approx 0.06081.
$$

Now compute the denominator:

$$
1 + 5 \times 0.06081 \approx 1 + 0.30405 = 1.30405.
$$

Then,

$$
P(7) \approx \frac{800}{1.30405} \approx 613.34.
$$

Rounding to the nearest whole number:

$$
\text{Population size after 7 years} \approx 613.
$$

### Final Answers
- Initial population size: $$ 133 $$
- Population size after 7 years: $$ 613 $$
Initial population size: 133 Population size after 7 years: 613

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