A species of fish was added to a lake. The population size P (t) of this species can be modeled by the following function, where t 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 after 9 years and round your answer to the nearest whole number as necessary. The equation is: P(t) = 1800/1+6e^-0.34t

The population size \( P(t) \) of the species can be modeled by the given function:

\[
P(t) = \frac{1800}{1 + 6e^{-0.34t}}
\]

### 1. **Finding the Initial Population Size \( P(0) \)**

The initial population size corresponds to \( t = 0 \).

Substitute \( t = 0 \) into the equation:

\[
P(0) = \frac{1800}{1 + 6e^{-0.34(0)}}
\]

Since \( e^0 = 1 \):

\[
P(0) = \frac{1800}{1 + 6 \times 1} = \frac{1800}{1 + 6} = \frac{1800}{7} \approx 257.14
\]

So, the initial population size is approximately **257** fish.

### 2. **Finding the Population After 9 Years \( P(9) \)**

Substitute \( t = 9 \) into the equation:

\[
P(9) = \frac{1800}{1 + 6e^{-0.34(9)}}
\]

Now, calculate \( -0.34 \times 9 \) and the exponential:

\[
P(9) = \frac{1800}{1 + 6e^{-3.06}}
\]

Calculate \( e^{-3.06} \) and substitute it back:

\[
P(9) = \frac{1800}{1 + 6 \times e^{-3.06}}
\]

\[
e^{-3.06} \approx 0.047
\]

\[
P(9) = \frac{1800}{1 + 6 \times 0.047} = \frac{1800}{1 + 0.282} = \frac{1800}{1.282} \approx 1403.43
\]

So, the population after 9 years is approximately **1403** fish.

### Summary:
- **Initial Population**: Approximately **257** fish.
- **Population After 9 Years**: Approximately **1403** fish.

Would you like any further details or have any questions?

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**Related Questions:**
1. How does the population growth rate change over time?
2. What happens to the population as \( t \) approaches infinity?
3. Can you derive the time at which the population will reach half of its carrying capacity?
4. What is the maximum population the lake can sustain based on this model?
5. How would changes in the growth rate constant affect the population over time?

**Tip:** Understanding the exponential function \( e^x \) is crucial for solving differential equations and analyzing population dynamics.

Explore how to model population growth using an exponential function. Find the initial population size and predict the population after 9 years using a given equation. Suitable for high school students learning about exponential functions and population dynamics.

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