where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants.(a) When would half the population have heard the rumor?(b) When is the rate of spread of the rumor greatest? **Understanding the Spread of a Rumor: The Logistic Growth Model**In this educational section, we explore how the spread of a rumor can be modeled mathematically. The equation used to describe this phenomenon is known as the logistic growth model.**The Logistic Growth Model Equation**The spread of a rumor over time $ t $ is modeled by the equation:\[ p(t) = \frac{1}{1 + ae^{-kt}} \]Where:- $ p(t) $ represents the proportion of the population that has heard the rumor at time $ t $.- $ a $ and $ k $ are positive constants that affect the shape and rate of the spread.### Detailed Explanation- **Proportion of Population $ p(t) $**: This function gives the fraction of the population that knows about the rumor at any given time $ t $. When $ t = 0 $, the proportion is minimal. As time progresses, $ p(t) $ increases, approaching 1, meaning the entire population eventually hears the rumor.- **Constants $ a $ and $ k $**: - **$ a $**: Determines the initial proportion of the population aware of the rumor. A larger $ a $ means the rumor spreads more slowly initially. - **$ k $**: The growth rate of the spread. Larger values of $ k $ lead to a quicker spread of the rumor.### Graphical RepresentationWhile no graph is provided in the original content, typically, a logistic growth curve looks like an "S" shape (sigmoidal curve) when plotted. Initially, the curve starts slowly, then rapidly rises in the middle phase, and finally levels off as it approaches the maximum value of 1.### Real-Life ApplicationUnderstanding this model helps in studying how quickly information, whether true or false, disseminates through social networks, aiding in better designing communication strategies in both marketing and public health to either promote or curb information spread.

Answered: Image /qna-images/answer/64806668-662e-47e4-b8d9-58a4c84dc87c.jpg

where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants. (a) When would half the population have heard the rumor? (b) When is the rate of spread of the rumor greatest?

where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants. (a) When would half the population have heard the rumor? (b) When is the rate of spread of the rumor greatest?

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants.

(a) When would half the population have heard the rumor?

(b) When is the rate of spread of the rumor greatest?

**Understanding the Spread of a Rumor: The Logistic Growth Model**

In this educational section, we explore how the spread of a rumor can be modeled mathematically. The equation used to describe this phenomenon is known as the logistic growth model.

**The Logistic Growth Model Equation**

The spread of a rumor over time $ t $ is modeled by the equation:

\[ p(t) = \frac{1}{1 + ae^{-kt}} \]

Where:
- $ p(t) $ represents the proportion of the population that has heard the rumor at time $ t $.
- $ a $ and $ k $ are positive constants that affect the shape and rate of the spread.

### Detailed Explanation

- **Proportion of Population $ p(t) $**: This function gives the fraction of the population that knows about the rumor at any given time $ t $. When $ t = 0 $, the proportion is minimal. As time progresses, $ p(t) $ increases, approaching 1, meaning the entire population eventually hears the rumor.

- **Constants $ a $ and $ k $**:
- **$ a $**: Determines the initial proportion of the population aware of the rumor. A larger $ a $ means the rumor spreads more slowly initially.
- **$ k $**: The growth rate of the spread. Larger values of $ k $ lead to a quicker spread of the rumor.

### Graphical Representation

While no graph is provided in the original content, typically, a logistic growth curve looks like an "S" shape (sigmoidal curve) when plotted. Initially, the curve starts slowly, then rapidly rises in the middle phase, and finally levels off as it approaches the maximum value of 1.

### Real-Life Application

Understanding this model helps in studying how quickly information, whether true or false, disseminates through social networks, aiding in better designing communication strategies in both marketing and public health to either promote or curb information spread.

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