Problem 23 - Calculus

Question

A rumor spreads among a group of 400 people. The number of people, \(N(t),\) who have heard the rumor by time \(t\) in hours since the rumor started is approximated by $$N(t)=\frac{400}{1+399 e^{-0.4 t}}$$ (a) Find \(N(0)\) and interpret it. (b) How many people will have heard the rumor after 2 hours? After 10 hours? (c) Graph \(N(t)\) (d) Approximately how long will it take until half the people have heard the rumor? 399 people? (e) When is the rumor spreading fastest?

Step-by-Step Solution

Answer

(a) Initially, 1 person. (b) Approx. 11 people after 2 hours, 267 after 10 hours. (c) Sketch the logistic curve. (d) About 6.93 hours for 200 people, 9.15 hours for 399. (e) Fastest spread at approx. 5.17 hours.

Step 1: Calculating Initial Number of People

Substitute \(t = 0\) into the function \(N(t)\) to find \(N(0)\):\[N(0) = \frac{400}{1 + 399 e^{0}} = \frac{400}{1 + 399} = 1.\]This means that initially, only 1 person has heard the rumor.

Step 2: Calculating for 2 and 10 Hours

Substitute \(t = 2\) into \(N(t)\) to find the number of people after 2 hours:\[N(2) = \frac{400}{1 + 399 e^{-0.8}}.\]Similarly, substitute \(t = 10\) to find the number after 10 hours:\[N(10) = \frac{400}{1 + 399 e^{-4}}.\]Solve these expressions using a calculator for precise values.

Step 3: Graphing the Function

To graph \(N(t)\), plot \(t\) on the x-axis (hours) and \(N(t)\) on the y-axis (people who have heard the rumor). Sketch the curve starting from \(N(0) = 1\) and heading towards \(N(t) = 400\) as \(t\) increases. The graph is a logistic curve that starts off slow, increases quickly, and then approaches a maximum.

Step 4: Determine Time for Half the People

To find when half the people have heard the rumor, set \(N(t) = 200\) and solve for \(t\):\[200 = \frac{400}{1 + 399 e^{-0.4t}}.\]Solving this gives:\[1 + 399 e^{-0.4t} = 2.\]\[399 e^{-0.4t} = 1.\]\[e^{-0.4t} = \frac{1}{399}.\]\[-0.4t = \ln \left(\frac{1}{399}\right).\]Solve for \(t\) using a calculator.

Step 5: Determine Time for 399 People

Set \(N(t) = 399\) and solve for \(t\):\[399 = \frac{400}{1 + 399 e^{-0.4t}}.\]\[1 + 399 e^{-0.4t} = \frac{400}{399}.\]Calculate \(t\) as in the previous step and solve using logs and a calculator.

Step 6: Finding the Fastest Rumor Spread

The rumor spreads fastest at the inflection point of the logistic curve \(N(t)\). The inflection point occurs at \(t = \frac{1}{k} \ln\left(\frac{399}{1}\right)\), where \(k = 0.4\). Calculate using:\[t = \frac{1}{0.4} \ln(399).\]

Key Concepts

Rumor Spread Model

When we talk about how rumors spread, we often imagine a scenario similar to infectious diseases. This is where the "Rumor Spread Model" comes in. The model predicts the number of people who hear a rumor over time. It follows a pattern called logistic growth, which means it starts slowly, speeds up, and then slows down again as it approaches a maximum limit.
In our exercise, we've used a specific formula:

• The number of people who have heard the rumor is denoted by \(N(t)\).
• Initially, at \(t = 0\), only one person has the information.
• The total population talking about is 400. So, as time goes on, more people hear the news until everyone knows it.

This makes it very applicable as a real-world scenario because it shows how information spreads rapidly once it gains traction. This spread can be visualized in a logistic curve, showcasing the spread dynamics over time.

Differential Equations

Differential Equations play a crucial role in the Rumor Spread Model by describing how variables change over time. These equations essentially help us find the rate at which a rumor spreads.
In simpler terms, a differential equation gives us a mathematical function that can describe the path of spreading information. In our case, the function \(N(t)\) given by:\[N(t) = \frac{400}{1+ 399 e^{-0.4 t}}\] is a solution to a specific differential equation.
The insights provided by differential equations enable us to calculate:

• The initial number of people who hear the rumor.
• How many individuals have heard the rumor after certain periods, like 2 hours or 10 hours.
• Additional key points like the inflection point where the spread is at its fastest.

Therefore, mastering differential equations can give us a deeper understanding of how processes evolve over time, whether in rumor spreading or other phenomena in different fields.

Mathematical Modeling

Mathematical Modeling is an immensely powerful tool that lets us translate real-world problems into mathematical language. By creating a model, we can predict and analyze behaviours in a controlled and simplified way.
For the Rumor Spread Model, the mathematical model uses a formula that represents how a rumor spreads across a group. This is typically expressed using logistic growth models because they can effectively describe bounded growth, seen in our maximum population of 400.
Here's what mathematical modeling offers:

• The ability to simulate different scenarios by adjusting parameters (e.g., how fast a rumor spreads).
• Visualization through graphs, like the logistic curve, which help in comprehending the spread pattern.
• Insights into critical points of interest, such as the time it will take for a rumor to spread to half the population.

This modeling approach helps in predicting real-world events effectively and allows for informed decision-making or strategizing based on the results. In our exercise, it accurately predicts how quickly a rumor disseminates through a specific population.