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...
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F24fae4ed-b1b1-4171-bba8-4d29349ae5e5%2F64806668-662e-47e4-b8d9-58a4c84dc87c%2F0058z5_processed.png&w=3840&q=75)
Transcribed Image Text:**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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning