Problem 11 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Suppose that in a certain country the population grows at a rate proportional to itself with proportionality constant \(0.02 .\) Further suppose that due to a drought people are leaving the country at a constant rate of 1000 people per year. Let \(P=P(t)\) be the population of the country at time \(t\), where \(t\) is in years. Write a differential equation modeling the situation.

Step-by-Step Solution

Answer

The differential equation representing the change in population is \(P'(t) = 0.02P(t) - 1000.\)

Step 1: Understand the Constants

There are two constants in the situation. First, the growth rate of the population is proportional with a constant of \(0.02 .\) This will be represented as \(0.02P\), where \(P\) is the population. Second, people are leaving at a constant rate of \(1000\) people per year. This will be represented simply as \(1000.\)

Step 2: Formulate the differential equation

The population growth is represented by \(P'=0.02P\), where \(P'\) signifies the rate of change of the population, i.e., the derivative of \(P\) with respect to \(t.\) The loss of population due to people leaving is represented by \(1000\) and has to be subtracted from the growth rate. So, the rate of change of the population \(P'\) is the difference between the growth rate and the loss rate. This can be represented as \(P'(t) = 0.02P(t) - 1000.\)

Step 3: State the final differential equation

The resulting differential equation that models the population change in the country over time is \(P'(t) = 0.02P(t) - 1000.\)

Key Concepts

Population Growth

Population growth is a fascinating concept and a vital part of understanding how communities and nations evolve over time. At its heart, population growth refers to the changes in the size of a group of individuals sharing the same habitat. In our scenario, this group is the population of a certain country. Factors affecting population growth include:

• Birth rates - the number of newborns in a population
• Death rates - the number of deaths in the population
• Migration - movement of people into or out of the population

In mathematical models, such as the one provided in the exercise, population growth is often simplified and measured as a percentage change over time. It is denoted as a rate, which then describes how the population evolves. In this case, the population grows at a rate determined by a growth constant, denoted as 0.02, meaning each year the population increases by 2% of its size in the absence of other factors.

Proportional Growth

Proportional growth is a term used to describe a situation where a quantity increases in a manner where the growth rate is directly linked to its current size. This is also known as exponential growth when dealing with populations. For the given exercise:

• The concept of "proportional" growth implies that the larger the population, the greater the increase within a given period.
• The proportionality constant of 0.02 signifies that the population's derivative is directly 2% of the population size.

Mathematically, this can be represented by the equation: \[ P' = kP \] where:

• \( P' \) is the rate of change in population size,
• \( k \) is the proportionality constant (0.02 in this instance), indicating how the growth evolves over time.

Such models are suitable for understanding how isolated populations grow or decline without external factors considerably affecting them.

Constant Rate

In this exercise, the term "constant rate" refers specifically to a fixed number of people leaving the country each year, regardless of the current population size. A constant rate is crucial in assessing real-life scenarios as it ensures the inclusion of non-variable factors influencing a system:

• In our exercise, 1000 people move out each year, which does not change based on the total population size.
• Unlike proportional growth which changes with the population size, a constant rate typically involves a subtraction of constant values in differential equations.

Thus, the term constant rate helps differentiate factors that are steadfast and continuous over time, giving the mathematical equation a more realistic approach. The constant rate element helps to embody practical influences in a natural growth scenario.