   A population of fish in a pond is described by the logistic function P(t)=2101+e−0.54tPt=2101+e−0.54t, where P(t) is the number of fish in t years.

It seems like you're working with a logistic function that models the population of fish in a pond over time. The function provided is:

\[
P(t) = \frac{210}{1 + e^{-0.54t}}
\]

Here, \( P(t) \) represents the number of fish at time \( t \) years.

### Understanding the Logistic Function:
- **Carrying Capacity**: The carrying capacity is the maximum population size that the environment can sustain indefinitely. In the logistic function, it is represented by the numerator. In this case, the carrying capacity is 210 fish.
- **Growth Rate**: The parameter \( 0.54 \) represents the growth rate. It determines how quickly the population approaches the carrying capacity.
- **Initial Population**: The initial population can be found by evaluating \( P(0) \).

### Let's calculate a few key details:

1. **Initial Population** (\( P(0) \)):
   \[
   P(0) = \frac{210}{1 + e^{0}} = \frac{210}{1 + 1} = \frac{210}{2} = 105
   \]

   So, the initial population of fish is 105.

2. **Population after 5 years** (\( P(5) \)):
   \[
   P(5) = \frac{210}{1 + e^{-0.54 \times 5}} = \frac{210}{1 + e^{-2.7}}
   \]
   We can calculate this to find the exact population after 5 years.

Would you like me to calculate the population at specific time points or explain any other details about the function?

### Related Questions:

1. How does the population change after 10 years?
2. What is the time \( t \) when the population reaches 150 fish?
3. How does the growth rate \( 0.54 \) affect the population size over time?
4. What happens to the population as \( t \) approaches infinity?
5. How would changing the carrying capacity (from 210 to another value) affect the population model?

**Tip**: In logistic growth models, the population grows rapidly at first and then slows down as it approaches the carrying capacity. This is typical in environments with limited resources.

Explore how a logistic function models the population of fish in a pond over time. Learn about the carrying capacity, growth rate, and initial population. Calculations include finding initial population and predicting population after specific years.

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