Problem 19 - Calculus Concepts

Question

Postage Stamps In 1880,37 countries issued postage stamps. The rate of change (with respect to time) of the number of countries issuing postage stamps between 1836 and 1880 was jointly proportional to the number of countries that had already issued postage stamps and to the number of countries that had not yet issued postage stamps. The constant of proportionality was approximately 0.0049. By 1855,16 countries had issued postage stamps. (Source: "The Curve of Cultural Diffusion," American Sociological Review, August \(1936,\) pp. \(547-556)\) a. Write a differential equation describing the rate of change in the number of countries issuing postage stamps with respect to the number of years since 1800 b. Write a general solution for the differential equation. c. Write the particular solution for the differential equation. d. Estimate the number of countries that were issuing postage stamps in 1840 and in \(1860 .\)

Step-by-Step Solution

Answer

The differential equation is \( \frac{dN}{dt} = 0.0049N(t)(37-N(t)) \). Solve for \( N(t) \) with initial conditions for a particular solution, then substitute \( t = 40, 60 \) for estimates.

Step 1: Understand the Problem Statement

We are given a problem about the rate of change of countries issuing postage stamps. The rate of change is jointly proportional to the number of countries that have issued and the number of countries that have not issued stamps. The proportionality constant is given as 0.0049. Initially, in 1855, 16 countries had issued stamps.

Step 2: Define the Variables

Let \( t \) be the years since 1800, \( N(t) \) be the number of countries that issued stamps at time \( t \), and \( C \) be the total number of countries (37 by 1880). Then the rate of change \( \frac{dN}{dt} \) is proportional to both \( N(t) \) and \( C - N(t) \).

Step 3: Formulate the Differential Equation

According to the given information, \( \frac{dN}{dt} = k \times N(t) \times (C-N(t)) \) where \( k = 0.0049 \) and \( C = 37 \). So the differential equation is: \[ \frac{dN}{dt} = 0.0049 \times N(t) \times (37 - N(t)) \]

Step 4: Solve the Differential Equation

This is a separable differential equation. Rearrange terms: \[ \frac{1}{N(37-N)} \frac{dN}{dt} = 0.0049 \] Integrating, we get \[ \int \frac{1}{N(37-N)} dN = 0.0049 \int dt \]. Solve the integrals to find \( N(t) \).

Step 5: Apply Initial Conditions for the Particular Solution

In 1855 (\( t = 55 \)), 16 countries had issued stamps. Use the initial condition \( N(55) = 16 \) to find the particular solution of \( N(t) \). Substitute \( t = 55 \) into the integrated equation and solve for constants of integration.

Step 6: Estimate Number of Countries in 1840 and 1860

Use the particular solution \( N(t) \) to estimate \( N(40) \) for 1840 and \( N(60) \) for 1860 by substituting these values of \( t \) into your particular solution equation.

Key Concepts

Rate of Change

When dealing with differential equations, the term "rate of change" is very important. In simple terms, it describes how a certain quantity, like the number of countries issuing postage stamps, changes over time.
In mathematical terms, this is often represented as \( \frac{dN}{dt} \), where \( N \) is the quantity of interest (in this case, the number of countries).
The rate of change gives us a way to predict how the number will change as time progresses. In our exercise, the rate of change depends on two factors: the number of countries that have already issued stamps and those that have not. This dual dependence is what characterizes the system as being jointly proportional, which we'll dive into next.
Understanding the rate at which the change happens helps to model real-world scenarios, like the spread of an idea or technology across different places.

Proportionality Constant

In various mathematical models, especially in differential equations, the proportionality constant plays a crucial role. It links the rate of change to influencing factors in the model.
In our exercise, the proportionality constant is given as 0.0049. This means the rate at which countries adopt postage stamps increases with both the number of countries that have already adopted it and the number of countries that haven't yet adopted it. The constant scales the impact of these two numbers on the rate.
The interpretation here is straightforward. A larger proportionality constant would mean a quicker adoption rate, while a smaller one would imply a slower adoption. Recognizing and applying the proportionality constant can help predict how rapidly changes occur in systems modeled by differential equations.

Cultural Diffusion

Cultural diffusion is the term used to describe the spread of cultural beliefs and social activities from one group to another. In the context of our exercise, the adoption of postage stamps is an example of cultural diffusion.
As one country adopts postage stamps, it influences others to do the same. This process is rarely random. Instead, it follows patterns that can often be described using differential equations. The spread depends on how connected and influential the initial adopters are to those who have not yet adopted the innovation.
In mathematical modeling, like our exercise, understanding cultural diffusion helps frame how innovations like postage stamps spread over time. Ultimately, studying this can give insight into how quickly or slowly new ideas or technologies take hold in different parts of the world.

Initial Conditions

In differential equations, initial conditions are the specific values provided that help solve for unknown constants in the solutions. They are crucial because they personalize and specify a general solution to fit the unique scenario we investigate.
In the exercise, the initial condition provided is that by 1855, 16 countries had issued postage stamps. The moment we plug this information into our equation, it allows us to find the constant that tailors the general solution to the problem's precise situation.
This means we can predict past or future adoption rates based on known data points, like the situation in 1855. Initial conditions are akin to setting the starting point on a map before embarking on a journey, ensuring our calculations are accurately tailored to the real world.