he equation  N(t)=550/(1+49e^(−0.7t))

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?

N(t):         approaches

The given equation for the number of people who have heard the rumor is:

\[
N(t) = \frac{550}{1 + 49e^{-0.7t}}
\]

### Step 1: Determine the behavior of \( N(t) \) as \( t \) increases without bound (i.e., as \( t \to \infty \)).

As \( t \to \infty \), the exponential term \( e^{-0.7t} \) tends to 0 because exponential decay causes \( e^{-kt} \) to shrink towards zero for positive \( k \) as \( t \) increases. Thus:

\[
\lim_{t \to \infty} e^{-0.7t} = 0
\]

Substituting this into the equation:

\[
\lim_{t \to \infty} N(t) = \frac{550}{1 + 49(0)} = \frac{550}{1} = 550
\]

Thus, **as \( t \to \infty \), \( N(t) \) approaches 550**. This means that the total number of people who will eventually hear the rumor is 550.

### Step 2: How many people started the rumor?

To find how many people started the rumor, substitute \( t = 0 \) into the equation:

\[
N(0) = \frac{550}{1 + 49e^{-0.7(0)}} = \frac{550}{1 + 49 \cdot 1} = \frac{550}{50} = 11
\]

So, **11 people started the rumor**.

### Interpretation:
- As time passes, eventually **550 people** will hear the rumor.
- Initially, **11 people** started the rumor.

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Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. What is the rate of change of the number of people hearing the rumor at \( t = 5 \) days?
2. How long will it take for half the population (275 people) to hear the rumor?
3. How does the number of people hearing the rumor behave between \( t = 0 \) and \( t = 10 \)?
4. What happens to the population hearing the rumor if the rate \( 0.7 \) is changed to a different value?
5. Can you find the inflection point where the number of people hearing the rumor grows fastest?

**Tip**: When dealing with exponential growth or decay models, the long-term behavior is determined by the exponential term, which tends to either 0 or infinity as time increases.

The equation N(t)=550/(1+49e^(−0.7t)) 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?

Explore how the exponential decay model N(t) = 550/(1+49e^(-0.7t)) predicts the spread of a rumor over time. Learn the long-term behavior of the model as N(t) approaches 550 and discover how 11 people initially started the rumor. Suitable for high school students studying exponential models.

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