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The image shows a logistic growth problem related to the fish population in a pond, governed by the logistic growth model:

\[
P(t) = \frac{d}{1 + ke^{-ct}}
\]
where \( P(t) \) represents the population at time \( t \), and \( d, k, \) and \( c \) are constants. Specifically, the constants provided are:

- \( d = 1400 \)
- \( k = 13 \)
- \( c = 0.2 \)
- \( t \) is measured in years.

Let's break down and answer the parts of the question:

### Part (a): How many fish were originally put in the pond?

This asks for the population at \( t = 0 \). To find it, we substitute \( t = 0 \) into the logistic model.

\[
P(0) = \frac{d}{1 + ke^{-ct}} = \frac{1400}{1 + 13 \cdot e^{-0.2 \cdot 0}} = \frac{1400}{1 + 13} = \frac{1400}{14} = 100
\]

Thus, 100 fish were originally put in the pond.

### Part (b): Find the population after 10, 20, and 30 years

We can calculate \( P(t) \) for \( t = 10, 20, 30 \). Let’s calculate these values step by step using the logistic equation.

I'll now compute the values for you.The population of fish after each time period is as follows:

- After 10 years: 507 fish
- After 20 years: 1131 fish
- After 30 years: 1356 fish

### Part (c): Evaluate \( P(t) \) for large values of \( t \). What value does the population approach as \( t \to \infty \)?

As \( t \) increases towards infinity, the exponential term \( e^{-ct} \) approaches zero, so the logistic equation simplifies to:

\[
P(\infty) = \frac{d}{1 + 0} = d = 1400
\]

Thus, the population approaches 1400 fish as time goes to infinity.

The graph shown in the problem confirms this behavior, with the population leveling off near 1400.

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Would you like any further details or clarifications? Here are five related questions to extend your understanding:

1. How does the logistic growth model differ from exponential growth?
2. What factors could change the maximum population capacity in a real-world scenario?
3. How would the model change if the initial population were larger?
4. How does the constant \( c \) affect the growth rate in the model?
5. How would the population evolve if \( k \) were smaller or larger?

**Tip:** In logistic growth models, the constant \( d \) represents the carrying capacity, the maximum population the environment can sustain.

Determine the initial fish population and the population after 10, 20, and 30 years using the logistic growth model formula P(t) = d / (1 + ke^(-ct)). Evaluate P(t) for large values of t.

Explore the dynamics of a fish population in a pond modeled by the logistic growth equation P(t) = d / (1 + ke^(-ct)), with parameters d = 1400, k = 13, and c = 0.2. Learn how to calculate initial population and future growth after 10, 20, and 30 years, and understand the behavior as time approaches infinity.

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