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.

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.