Problem 80 - Elementary and Intermediate Algebra

Question

The population of Earth is approximately 6.8 billion people and is growing at an annual rate of \(1.133 \% .\) Use the exponential growth model to predict the world population in 30 years.

Step-by-Step Solution

Answer

The predicted world population in 30 years is approximately 9.68 billion.

Step 1: Understanding the Exponential Growth Model

The exponential growth model allows us to predict future population sizes based on current population and growth rate. It uses the formula: \[ P(t) = P_0 imes (1 + r)^t \]where:- \( P(t) \) is the population at time \( t \).- \( P_0 \) is the initial population size.- \( r \) is the annual growth rate in decimal form.- \( t \) is the number of years into the future we want to predict.

Step 2: Identify Initial Values

Let's identify the values from the problem:- Initial population size, \( P_0 = 6.8 \) billion.- Annual growth rate, \( r = 1.133 \% = 0.01133 \) (as a decimal).- Time, \( t = 30 \) years.

Step 3: Plug Values into the Formula

Insert the identified values into the exponential growth formula:\[ P(30) = 6.8 imes (1 + 0.01133)^{30} \]

Step 4: Calculate the Base of the Exponent

Calculate \( (1 + 0.01133) \):\[ 1 + 0.01133 = 1.01133 \]

Step 5: Compute the Exponent

Calculate \( (1.01133)^{30} \):- Using a calculator, \( (1.01133)^{30} \approx 1.423416 \).

Step 6: Final Calculation of Future Population

Multiply 6.8 billion by the calculated exponent:\[ P(30) = 6.8 imes 1.423416 \approx 9.678429 \] billion.

Step 7: Conclusion

Thus, the predicted world population in 30 years is approximately 9.68 billion.

Key Concepts

Understanding Population Prediction

Population prediction involves estimating the future size of a population based on current data. This process is essential for planning resources, understanding trends, and making informed decisions about the future.
To predict population, you need:

• A reliable initial population size.
• An accurate growth rate.
• A clear understanding of the time period for prediction.

In our exercise, the goal is to forecast the world's population in 30 years. This requires applying these elements into a formula, revealing how an initially small percentage growth can translate into millions more people over time.
Such predictions help governments, organizations, and scientists address future challenges, like infrastructure needs or environmental impacts.

Decoding Growth Rate

The growth rate is key to understanding how rapidly a population increases. It indicates the percentage change in population size over a specific period, often annually.
The rate given for the Earth's population is 1.133%. For calculations, it is converted to a decimal. This involves moving the decimal place two steps left:

• 1.133% becomes 0.01133.

This rate, although seemingly small, has a compound effect when used over many years. It contributes to a significant increase in population size, as a consistent percentage of growth compounds upon itself annually.
This concept emphasizes why small growth rates can lead to very large populations over time. It highlights the momentum inherent in exponential growth processes.

Exploring the Exponential Growth Model

The exponential growth model is a mathematical tool designed for predicting how a population evolves over time given a constant growth rate. This model relies on the principle that the growth rate applies not just to the original population but also compounds on the new additional population each year.
The formula used is: \[ P(t) = P_0 \times (1 + r)^t \] Where:

• \( P(t) \) represents the population at time \( t \).
• \( P_0 \) is the starting population size.
• \( r \) is the growth rate in decimal form.
• \( t \) is the time period being considered.

In the given problem, plugging in the numbers shows how this formula accurately predicts a 30-year future population based on known starting points.
This model is highly effective for situations with stable growth rates, offering simplicity and precision in forecasting. Such a tool becomes invaluable when projecting trends in ecology, economics, and demography.