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Let's solve this problem step by step:

### Part (a): 
We are given that the fish population follows a continuous exponential growth model, meaning it can be modeled using the formula:
\[
y = y_0 e^{kt}
\]
where:
- \( y \) is the fish population at time \( t \) years,
- \( y_0 \) is the initial population (340 fish),
- \( k \) is the growth rate,
- \( t \) is the time in years.

The problem tells us:
- Initial population \( y_0 = 340 \),
- After 5 years (\( t = 5 \)), the population is 544 (\( y(5) = 544 \)).

We can use this information to find \( k \), the growth rate.

The equation at \( t = 5 \) is:
\[
544 = 340 e^{5k}
\]

To solve for \( k \), we first divide both sides by 340:
\[
\frac{544}{340} = e^{5k}
\]

Taking the natural logarithm (ln) of both sides:
\[
\ln\left(\frac{544}{340}\right) = 5k
\]

Now we solve for \( k \):
\[
k = \frac{1}{5} \ln\left(\frac{544}{340}\right)
\]

### Part (b):
Once we have \( k \), we can predict the population 22 years after the initial population is introduced. This means we need to find \( y(22) \) using the formula:
\[
y = 340 e^{k \cdot 22}
\]

Let me calculate the values.The growth rate \( k \) is approximately \( 0.094 \).

### Part (a): The formula for the fish population \( y \) as a function of time \( t \) is:
\[
y = 340 e^{0.094t}
\]

### Part (b): After 22 years, the fish population will be approximately **2,689** fish.

Would you like further explanation or have any questions?

Here are 5 related questions to deepen your understanding:
1. How does the exponential growth model differ from a linear growth model?
2. What would happen to the population if the growth rate \( k \) were negative?
3. How can you derive the doubling time for this population?
4. How does the initial population size affect the population after a long time?
5. Can you modify this model to include a carrying capacity for the lake?

**Tip:** In exponential models, small changes in the growth rate \( k \) can have a large impact over time due to the nature of exponential growth.

An initial population of 340 fish is introduced into a lake. The fish population grows according to a continuous exponential growth model. There are 544 fish in the lake after 5 years. Write a formula relating y to t, and calculate the population after 22 years.

Solve the exponential growth model for a fish population, starting with 340 fish and growing to 544 after 5 years. Learn how to calculate the growth rate and predict future population after 22 years using continuous exponential growth.

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