Solving for the Number of People

Oct 13, 2025 by wbrum.org 115 views

Introduction

When an event happens in a community, news travels fast. It's not uncommon to see how quickly information can spread from person to person. In this article, we'll explore a mathematical model to describe the spread of news in a community. We'll assume that 6 people initially know the news and the number of people who hear the news increases by a factor of 4.6 each hour. Our goal is to find a formula that gives the number of people who hear the news as a function of time.

The Basic Idea

The basic idea behind this model is that each person who hears the news will, in turn, tell 4.6 other people about it. This creates a chain reaction, where the number of people who hear the news increases exponentially with time. To model this, we can use the concept of exponential growth.

Exponential Growth

Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the more people who hear the news, the faster the number of people who hear the news will increase. Mathematically, this can be represented by the equation:

N(t) = N0 * (1 + r)^t

where:

N(t) is the number of people who hear the news at time t
N0 is the initial number of people who know the news (6 in this case)
r is the growth rate (4.6 in this case)
t is the time in hours

The Formula

However, the formula above assumes a constant growth rate, which may not be the case in reality. In our model, the growth rate increases by a factor of 4.6 each hour. To account for this, we can use the formula:

N(t) = N0 * (4.6)^t

This formula represents the number of people who hear the news as a function of time, taking into account the increasing growth rate.

Derivation of the Formula

To derive the formula, we can start with the basic idea that each person who hears the news will tell 4.6 other people about it. This creates a chain reaction, where the number of people who hear the news increases exponentially with time.

Let's assume that at time t, there are N(t) people who hear the news. Then, at time t+1, each of these people will tell 4.6 other people about it, resulting in a total of 4.6N(t) new people who hear the news. The total number of people who hear the news at time t+1 is then:

N(t+1) = N(t) + 4.6N(t) = (1 + 4.6)N(t) = 5.6N(t)

This process can be repeated for each subsequent hour, resulting in the formula:

N(t) = N0 * (5.6)^t

However, this formula assumes a constant growth rate of 5.6, which is not the case in our model. To account for the increasing growth rate, we can use the formula:

N(t) = N0 * (4.6)^t

Solving for the Number of People

Now that we have the formula, we can solve for the number of people who hear the news at any given time. Let's say we want to find the number of people who hear the news after 5 hours. We can plug in the values into the formula:

N(5) = 6 * (4.6)^5 = 6 * 1,011,419.36 = 6,068,216.16

So, after 5 hours, approximately 6,068,216 people will have heard the news.

Conclusion

In this article, we've explored a mathematical model to describe the spread of news in a community. We've assumed that 6 people initially know the news and the number of people who hear the news increases by a factor of 4.6 each hour. Our goal was to find a formula that gives the number of people who hear the news as a function of time. We've derived the formula:

N(t) = N0 * (4.6)^t

and used it to solve for the number of people who hear the news after 5 hours. This formula can be used to model the spread of news in a community and can be applied to a wide range of real-world scenarios.

References

[1] Wikipedia. (2023). Exponential growth. Retrieved from https://en.wikipedia.org/wiki/Exponential_growth
[2] Khan Academy. (2023). Exponential growth and decay. Retrieved from https://www.khanacademy.org/math/differential-equations/first-order-linear-differential-equations/exponential-growth-decay/v/exponential-growth-decay

Mathematical Model

Assumptions

6 people initially know the news
The number of people who hear the news increases by a factor of 4.6 each hour

Variables

N(t) = number of people who hear the news at time t
N0 = initial number of people who know the news (6)
r = growth rate (4.6)
t = time in hours

Formula

N(t) = N0 * (4.6)^t

Derivation

N(t+1) = N(t) + 4.6N(t) = (1 + 4.6)N(t) = 5.6N(t)

N(t) = N0 * (5.6)^t

However, this formula assumes a constant growth rate of 5.6, which is not the case in our model. To account for the increasing growth rate, we can use the formula:

N(t) = N0 * (4.6)^t

Solution

N(5) = 6 * (4.6)^5 = 6 * 1,011,419.36 = 6,068,216.16

Introduction

In our previous article, we explored a mathematical model to describe the spread of news in a community. We assumed that 6 people initially know the news and the number of people who hear the news increases by a factor of 4.6 each hour. We derived a formula to give the number of people who hear the news as a function of time. In this article, we'll answer some frequently asked questions about the spread of news in a community.

Q: What is the initial number of people who know the news?

A: The initial number of people who know the news is 6.

Q: What is the growth rate of the number of people who hear the news?

A: The growth rate of the number of people who hear the news is 4.6 each hour.

**Q: How does the formula N(t) = N0 * (4.6)^t work?**

A: The formula N(t) = N0 * (4.6)^t represents the number of people who hear the news at time t. The initial number of people who know the news (N0) is multiplied by the growth rate (4.6) raised to the power of t, where t is the time in hours.

Q: Can I use this formula to model the spread of news in a community with a different initial number of people who know the news?

A: Yes, you can use this formula to model the spread of news in a community with a different initial number of people who know the news. Simply replace N0 with the initial number of people who know the news in your community.

Q: Can I use this formula to model the spread of news in a community with a different growth rate?

A: Yes, you can use this formula to model the spread of news in a community with a different growth rate. Simply replace 4.6 with the growth rate in your community.

Q: How accurate is this formula?

A: The accuracy of this formula depends on the assumptions made in the model. If the assumptions are accurate, the formula should give a good estimate of the number of people who hear the news. However, if the assumptions are not accurate, the formula may not give a good estimate.

Q: Can I use this formula to model the spread of news in a community with a different time unit?

A: Yes, you can use this formula to model the spread of news in a community with a different time unit. Simply replace t with the time unit in your community.

Q: Can I use this formula to model the spread of news in a community with a different population size?

A: Yes, you can use this formula to model the spread of news in a community with a different population size. Simply replace N0 with the population size in your community.

Q: What are some limitations of this formula?

A: Some limitations of this formula include:

The formula assumes that the growth rate is constant, which may not be the case in reality.
The formula assumes that the initial number of people who know the news is 6, which may not be the case in reality.
The formula assumes that the time unit is hours, which may not be the case in reality.
The formula assumes that the population size is constant, which may not be the case in reality.

Conclusion

In this article, we've answered some frequently asked questions about the spread of news in a community. We've discussed the initial number of people who know the news, the growth rate, and the formula N(t) = N0 * (4.6)^t. We've also discussed some limitations of this formula and how it can be used to model the spread of news in a community with different initial conditions.

References

[1] Wikipedia. (2023). Exponential growth. Retrieved from https://en.wikipedia.org/wiki/Exponential_growth
[2] Khan Academy. (2023). Exponential growth and decay. Retrieved from https://www.khanacademy.org/math/differential-equations/first-order-linear-differential-equations/exponential-growth-decay/v/exponential-growth-decay

Mathematical Model

Assumptions

6 people initially know the news
The number of people who hear the news increases by a factor of 4.6 each hour

Variables

N(t) = number of people who hear the news at time t
N0 = initial number of people who know the news (6)
r = growth rate (4.6)
t = time in hours

Formula

N(t) = N0 * (4.6)^t

Derivation

N(t+1) = N(t) + 4.6N(t) = (1 + 4.6)N(t) = 5.6N(t)

N(t) = N0 * (5.6)^t

However, this formula assumes a constant growth rate of 5.6, which is not the case in our model. To account for the increasing growth rate, we can use the formula:

N(t) = N0 * (4.6)^t

Solution

N(5) = 6 * (4.6)^5 = 6 * 1,011,419.36 = 6,068,216.16

So, after 5 hours, approximately 6,068,216 people will have heard the news.