Problem 14 - Calculus

Question

(a) Define the variables. (b) Write a differential equation to describe the relationship. (c) Solve the differential equation. In \(2010,\) the population of India was 1.15 billion people and increasing at a rate proportional to its population. If the population is measured in billions of people and time is measured in years, the constant of proportionality is 0.0135.

Step-by-Step Solution

Answer

The population model is \( P(t) = 1.15e^{0.0135t} \).

Step 1: Define Variables

Let's define the variables to model the population growth of India. Let \( P(t) \) represent the population in billions of people at time \( t \), where \( t \) is the number of years after 2010. Therefore, at \( t = 0 \), \( P(0) = 1.15 \) billion.

Step 2: Construct the Differential Equation

The population growth rate is proportional to the current population. Therefore, we can express this relationship with the differential equation \( \frac{dP}{dt} = kP \), where \( k \) is the constant of proportionality, given as 0.0135.

Step 3: Solve the Differential Equation

We solve the differential equation \( \frac{dP}{dt} = 0.0135P \) using separation of variables. First, separate the variables: \( \frac{1}{P} \, dP = 0.0135 \, dt \). Integrate both sides: \( \int \frac{1}{P} \, dP = \int 0.0135 \, dt \). This results in \( \ln |P| = 0.0135t + C \). Exponentiate both sides to solve for \( P \): \( P = e^{0.0135t + C} = e^C \cdot e^{0.0135t} \). Let \( A = e^C \). Thus, \( P(t) = Ae^{0.0135t} \).

Step 4: Determine the Constant A

Use the initial condition to determine \( A \). At \( t = 0 \), \( P(0) = 1.15 \). Substitute into the equation: \( 1.15 = A \cdot e^{0.0135 \cdot 0} = A \). Therefore, \( A = 1.15 \). So, the function becomes \( P(t) = 1.15e^{0.0135t} \).

Key Concepts

Population Growth

Population growth describes the increase or decrease in the size of a population over time. It is often modeled in mathematics using differential equations. Why are differential equations crucial? Because they can describe how the rate of change of a population relates to the population itself. In our problem, we're looking at the population of India, which is growing over time.

• The population is measured in billions of people.
• Time is measured in years since 2010.

In mathematical terms, the population increases at a rate proportional to its current size. This means if you know the population's current size, you can predict how quickly it will grow.
This relationship is expressed with a differential equation, making it a powerful tool in understanding population trends.

Separation of Variables

Separation of variables is a technique for solving differential equations. It involves rearranging an equation so that each variable is on a different side. In our problem, we started with the differential equation \( \frac{dP}{dt} = kP \), where \( P \) is population and \( k \) is a constant.

Here's a step-by-step breakdown of how it works:

• First, we move the terms around to get \( \frac{1}{P} \, dP = k \, dt \), which places \( P \) and \( t \) on opposite sides of the equation.
• Next, we integrate both sides: \( \int \frac{1}{P} \, dP = \int k \, dt \).

After integrating, you'll find \( \ln |P| = kt + C \), where \( C \) is an integration constant.
Finally, we solve for \( P \) by exponentiating both sides to get \( P = e^{kt + C} = A \cdot e^{kt} \). This step-by-step method simplifies complex relationships into manageable parts.

Initial Condition

An initial condition is vital for finding a particular solution to a differential equation. It provides a specific starting point you can use to solve for an unknown constant. In our case, we need the initial condition to determine \( A \) in the equation \( P(t) = A e^{0.0135t} \).

With the problem for India's population, we know:

• In 2010 (\( t = 0 \)), the population \( P(0) = 1.15 \) billion.

By substituting this into our expression for \( P(t) \), we have \( 1.15 = A \).
This tells us that the constant \( A \), which determines our specific solution, is 1.15. Applying the initial condition pinpoints the unique behavior of the system from that starting point.