Problem 23 - Advanced Engineering Mathematics

Question

In March 1976, the world population reached 4 billion. A popular news magazine predicted that with an average yearly growth rate of \(1.8 \%\), the world population would be 8 billion in 45 years. How does this value compare with that predicted by the model that says the rate of increase is proportional to the population at any time \(t\) ?

Step-by-Step Solution

Answer

The projected population is greater with exponential growth (9.49 billion) than with continuous growth (9.18 billion).

Step 1: Understand the Problem

The current population at the start is 4 billion in 1976, with a predicted yearly growth rate of 1.8%. We need to compare the population in 2021 based on the compound interest formula vs. the continuous growth model.

Step 2: Formula for Exponential Growth

In exponential growth, population grows according to the formula: \( P = P_0 \times (1 + r)^t \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years.

Step 3: Calculate Population Using Exponential Growth

Using the formula: \( P = 4 \times (1 + 0.018)^{45} \). Calculate the future population: \( P = 4 \times (1.018)^{45} \approx 9.49 \text{ billion} \).

Step 4: Formula for Continuous Growth

For continuous growth, we use the formula \( P = P_0 \times e^{rt} \), where \( e \) is approximately 2.718. Here, \( r = 0.018 \) and \( t = 45 \).

Step 5: Calculate Population Using Continuous Growth

Apply the formula: \( P = 4 \times e^{(0.018 \times 45)} \). This results in \( P = 4 \times e^{0.81} \approx 9.18 \text{ billion} \).

Step 6: Compare Both Models

The population predicted by the exponential growth model is 9.49 billion, while the continuous growth model predicts 9.18 billion.

Key Concepts

Population Growth Model

Population growth is often studied using mathematical models to predict future population sizes. A popular method is the exponential growth model, which assumes that the rate of population increase is proportional to the current population size.
The formula for this model is given by:

• \( P = P_0 \times (1 + r)^t \)

Here, \( P_0 \) represents the initial population, \( r \) is the constant growth rate expressed as a decimal, and \( t \) is the time period in years.
For instance, a population of 4 billion with an annual growth rate of 1.8% over 45 years becomes:

• \[ P = 4 \times (1 + 0.018)^{45} \approx 9.49 \text{ billion} \]

Understanding this model helps to anticipate long-term growth based on current trends.

Continuous Growth Model

In many real-world scenarios, populations don't grow in distinct steps but continuously over time. The continuous growth model better reflects scenarios where change happens all the time, not at fixed intervals.
Using the continuous growth model, the population is given by the formula:

• \( P = P_0 \times e^{rt} \)

Here, \( e \) is the base of the natural logarithm, approximately equal to 2.718, \( r \) is the growth rate, \( t \) is the time in years, and \( P_0 \) is the starting population.
In our example:

• \[ P = 4 \times e^{0.018 \times 45} \approx 9.18 \text{ billion} \]

This model, often used for biological populations, accounts for a smoother, more continuous increase.

Compound Interest Formula

While the compound interest formula is traditionally used in finance, it is similar to the exponential growth model for populations. It calculates the future value of an investment based on periodic compounding.
The compound interest formula is:

• \( A = P \times (1 + \frac{r}{n})^{nt} \)

where \( A \) is the amount after time \( t \), \( P \) is the principal investment, \( r \) is the annual interest rate, and \( n \) is the number of times interest is compounded per year.
This formula highlights how exponential growth accumulates quickly and can be applied similarly in population growth if compounded periodically.
Even though we don't directly use this formula for populations, understanding the principle of compounding helps appreciate how small, consistent growth over time can lead to significant increases.