Math Problem Statement
Solution
The image presents a problem in which the number of fish in a pond, , is modeled by a piecewise function depending on time . The function has different definitions for three intervals of :
\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.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Piecewise Functions
Modeling with Functions
Formulas
Limit of (200 + 80t) / (2 + 0.05t) as t approaches infinity
Theorems
Limit properties for rational functions
Suitable Grade Level
Grades 11-12
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