A lake is stocked with 569 fish of a new variety. The size of the lake, the availability of food, and the number of other fish
restrict growth in the lake to a limiting value of 3,556 . The population of fish in the lake after time , in months, is given
by the function
.
Find the population after 1 months.

Ask by Crawford Robbins. in the United States

Mar 20,2025

Question

A lake is stocked with 569 fish of a new variety. The size of the lake, the availability of food, and the number of other fish
restrict growth in the lake to a limiting value of 3,556 . The population of fish in the lake after time , in months, is given
by the function
.
Find the population after 1 months.

Ask by Crawford Robbins. in the United States

Mar 20,2025

UpStudy AI · Accepted Answer

We are given the logistic population model

$$
P(t)=\frac{3556}{1+5.3e^{-0.27t}},
$$

and we need to find the population after $$ t = 1 $$ month. Substituting $$ t = 1 $$ into the model, we have

$$
P(1)=\frac{3556}{1+5.3e^{-0.27}}.
$$

**Step 1. Compute the exponential term**

Calculate

$$
e^{-0.27}.
$$

Using a calculator, we approximate:

$$
e^{-0.27}\approx 0.763.
$$

**Step 2. Multiply by 5.3**

Multiply this result by 5.3:

$$
5.3 \times 0.763 \approx 4.0439.
$$

**Step 3. Compute the denominator**

The denominator becomes

$$
1 + 4.0439 \approx 5.0439.
$$

**Step 4. Find $$ P(1) $$**

Substitute the denominator back into the formula:

$$
P(1) = \frac{3556}{5.0439} \approx 705.
$$

Thus, the population after 1 month is approximately $$705$$ fish.
After 1 month, there are approximately 705 fish in the lake.