The equation $$ N(t)=\frac{650}{1+49 e^{-0.7 t}} $$ models the number of people in a town who have heard a rumor after $$ t $$ days. As $$ t $$ increases without bound, what value does $$ N(t) $$ approach? Interpret your answer. How many people started the rumor? Number $$ N(t) $$ approaches Number $$ N(t) $$ is limited by the carrying capacity of the town. $$ N(t) $$ is limited by the number of poeple who started the rumor. $$ N(t) $$ is limited by the rate at which the rumor spreads. $$ { }^{3} N(t) $$ is limited by the number of days it takes for the entire population to hear the rumor. $$ N(t) $$ is not limited by any value and increases without bound.

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To analyze the given equation and address the questions, let's break it down step by step.

### Given Equation:
$$ N(t) = \frac{650}{1 + 49 e^{-0.7 t}} $$

### 1. **Limit of $$ N(t) $$ as $$ t $$ Approaches Infinity**

To find the value that $$ N(t) $$ approaches as $$ t $$ increases without bound:

$$
\lim_{t \to \infty} N(t) = \lim_{t \to \infty} \frac{650}{1 + 49 e^{-0.7 t}}
$$

As $$ t \to \infty $$:
- $$ e^{-0.7 t} $$ approaches **0** because the exponential decay causes the term to diminish.
  
Thus,

$$
\lim_{t \to \infty} N(t) = \frac{650}{1 + 0} = 650
$$

**Interpretation:**
- **$$ N(t) $$ approaches 650** as $$ t $$ becomes very large.
- This means that **eventually, 650 people in the town will have heard the rumor**. This value represents the **carrying capacity** of the rumor spread within the town.

### 2. **Number of People Who Started the Rumor**

To determine how many people started the rumor, evaluate $$ N(t) $$ at $$ t = 0 $$:

$$
N(0) = \frac{650}{1 + 49 e^{-0.7 \times 0}} = \frac{650}{1 + 49 \times 1} = \frac{650}{50} = 13
$$

**Interpretation:**
- **13 people** started the rumor initially.

### 3. **Choosing the Correct Interpretation**

Among the provided options, the most accurate interpretation based on the limit is:

- **"N(t) is limited by the carrying capacity of the town."**

This aligns with the mathematical result that $$ N(t) $$ approaches 650, indicating that the spread of the rumor is constrained by the total population size of the town.

### **Summary**

- **As $$ t $$ increases without bound, $$ N(t) $$ approaches **650**.**
- **13 people started the rumor.**
- **The spread of the rumor is limited by the carrying capacity of the town.**

Answer Summary:
As t becomes large, N(t) approaches 650, meaning ultimately 650 town residents hear the rumor. Additionally, 13 people started the rumor.
As $$ t $$ increases, $$ N(t) $$ approaches 650, meaning 650 people will eventually hear the rumor. The rumor was started by 13 people.

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Given Equation:

1. Limit of as Approaches Infinity

2. Number of People Who Started the Rumor

3. Choosing the Correct Interpretation

Summary

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