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The image presents a problem in which the number of fish in a pond, \(N(t)\), is modeled by a piecewise function depending on time \(t\). The function has different definitions for three intervals of \(t\):

\[
N(t) = 
\begin{cases} 
f(t) & 0 \leq t < 6 \\
25t + 150 & 6 \leq t < 8 \\
\frac{200 + 80t}{2 + 0.05t} & t \geq 8 
\end{cases}
\]
where \(f(t)\) is a continuous function, and we are also given that \(f(0) = 80\).

### Part (a) asks to:
1. **Find \(\lim_{t \to \infty} N(t)\)**.
2. **Explain the meaning of this limit in the context of the problem**.

### Step 1: Find \(\lim_{t \to \infty} N(t)\)
We are asked to compute the limit of \(N(t)\) as \(t\) approaches infinity. Since \(N(t)\) changes form for different intervals, the limit as \(t \to \infty\) applies to the third part of the piecewise function:

\[
N(t) = \frac{200 + 80t}{2 + 0.05t}, \text{ for } t \geq 8.
\]

We will compute the limit of this expression as \(t \to \infty\).

### Step 2: Explain the meaning of the limit
The limit will represent the long-term behavior of the fish population in the pond.

Let me first calculate the limit.The limit of \(N(t)\) as \(t \to \infty\) is 1600. This means that, over a long period of time, the number of fish in the pond will approach 1600. 

### Interpretation:
The value \( \lim_{t \to \infty} N(t) = 1600 \) suggests that, eventually, the fish population in the pond stabilizes at 1600 fish as time passes, implying a steady-state or equilibrium population size for the fish.

Do you need more details on the steps, or have any further questions?

Here are 5 related questions you can explore:
1. How would you calculate the limit of \( f(t) \) for \( t \to \infty \) if more information about \( f(t) \) was provided?
2. What conditions might lead to the fish population stabilizing at a certain number?
3. How does the form of the piecewise function relate to changes in the population growth rate over time?
4. What would happen if the growth rate for \( t \geq 8 \) changed from linear to exponential?
5. How would we model the fish population if environmental factors like food supply or space were limiting?

**Tip**: Always check for continuity at points where a piecewise function changes to ensure a smooth transition between intervals.

The number of fish in a pond at time t years is modeled by the function N(t) defined above, where f is a continuous function such that f(0) = 80. (a) Find lim(t -> ∞) N(t). Explain the meaning of lim(t -> ∞) N(t) in the context of the problem.

Explore how to calculate the limit of a piecewise function N(t) to model the population growth of fish in a pond. This problem uses limit properties of rational functions to explain long-term behavior. Perfect for high school students in Grades 11-12 studying calculus and real-world function modeling.

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