assume the growth rate of a certain species is constant but negative. sketch the population of at t=0, n=no what happens as t approaches infinity? is your answer expected?

Given that the growth rate of the species is constant but **negative**, the population of the species will decrease over time. This type of growth can be modeled using the **exponential decay** formula:

\[
N(t) = N_0 e^{rt}
\]

Where:
- \( N(t) \) is the population at time \( t \),
- \( N_0 \) is the initial population at \( t = 0 \),
- \( r \) is the growth rate (which is negative here),
- \( t \) is time.

### Behavior at \( t = 0 \)
At \( t = 0 \), the population is \( N(0) = N_0 \). This means the population starts at its initial value.

### Behavior as \( t \to \infty \)
Since the growth rate \( r \) is negative, as \( t \) increases, the exponential term \( e^{rt} \) decreases because \( e^{-\infty} \to 0 \). Thus, as \( t \to \infty \), the population \( N(t) \) will approach zero.

\[
\lim_{t \to \infty} N(t) = N_0 \cdot 0 = 0
\]

### Is this result expected?
Yes, this is expected because a negative growth rate implies that the population is shrinking over time. Over a long period, the population will approach zero, indicating the species could become extinct if the trend continues.

### Graphical Sketch:
- At \( t = 0 \), the population starts at \( N_0 \).
- The graph will show a **decreasing exponential curve** approaching zero as \( t \to \infty \).

If you want, I can generate an illustrative graph for this scenario.

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Do you need more details or have any questions?

### 5 Relative Questions:
1. How does a negative growth rate affect the species in the short term compared to the long term?
2. What would the equation look like if the growth rate were positive?
3. How does the value of \( r \) influence the speed at which the population decreases?
4. What would happen to the population if the growth rate were zero?
5. How does this model relate to real-world species facing environmental challenges?

### Tip:
In real-world scenarios, populations rarely follow a perfect exponential decay because other factors like resource availability, competition, and environmental changes can modify the growth rate over time.

Assume the growth rate of a certain species is constant but negative. Sketch the population of at t=0, n=no what happens as t approaches infinity? Is your answer expected?

Understand how to model the population of a species with a constant negative growth rate using exponential decay. Explore the behavior of the population at t=0 and as time approaches infinity. This problem includes the formula N(t) = N_0 * e^(rt) and analyzes the long-term effects of a negative growth rate. Ideal for students in Grades 10-12.

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