Problem 30 - Applied Calculus

Question

In 2004 , the world's population was \(6.4\) billion, and the population was projected to reach \(8.5\) billion by the year 2030 . What annual growth rate is projected?

Step-by-Step Solution

Answer

The projected annual growth rate is approximately 1.14%.

Step 1: Understanding Exponential Growth

To find the annual growth rate, we need to use the exponential growth model since population grows exponentially. The formula for exponential growth is given by \[ P(t) = P_0 (1 + r)^t \] where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years. Our task is to find \( r \).

Step 2: Assigning Known Values

Let's start by assigning the known values to our formula. We have the initial population in 2004, \( P_0 = 6.4 \) billion, and the projected population in 2030, \( P(t) = 8.5 \) billion. The time \( t \) is the difference between 2030 and 2004, so \( t = 26 \) years.

Step 3: Setting up the Equation

Substitute the known values into the exponential growth formula: \[ 8.5 = 6.4 (1 + r)^{26} \]. Our task now is to solve for \( r \), the annual growth rate.

Step 4: Isolating the Growth Factor

First, divide both sides of the equation by 6.4 to isolate the growth factor: \[ \frac{8.5}{6.4} = (1 + r)^{26} \]. Calculate \( \frac{8.5}{6.4} \) to get approximately 1.328125.

Step 5: Solving for the Growth Rate

To solve for \( r \), take the 26th root of 1.328125: \[ 1 + r = (1.328125)^{\frac{1}{26}} \]. Compute \( (1.328125)^{\frac{1}{26}} \) which gives approximately 1.0114.

Step 6: Finding the Growth Rate

Now solve for \( r \): \[ r = 1.0114 - 1 \] which simplifies to \( r \approx 0.0114 \).

Step 7: Converting to Percentage

To express \( r \) as a percentage, multiply by 100: \[ 0.0114 \times 100 = 1.14\% \]. The projected annual growth rate is approximately 1.14%.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population over a specified time period. It is a fundamental concept in demography and ecology. Understanding how populations grow is essential for planning and sustainability, as it affects various aspects of society such as resources, environment, and infrastructure.
Factors influencing population growth include:

• Birth rates
• Death rates
• Immigration
• Emigration

When a population grows without any restrictions, it can lead to exponential growth. This means the population size increases rapidly over time, as each generation adds more individuals than the previous one.
Hence, predicting future population sizes accurately is crucial for anticipating future needs and challenges.

Annual Growth Rate

The annual growth rate is a measure that allows us to understand how fast a population is increasing within a year. It is usually expressed as a percentage. Calculating this rate helps in assessing the pace of change in population size, which in turn can inform policy decisions and economic planning.
To find the annual growth rate, it is often necessary to use models like the exponential growth model. Once determined, this rate offers insights into whether population controls or supportive measures are needed.
The annual growth rate can be derived from the equation:

• By using the future population size, initial population size, and the time period in years.

Exponential Growth Model

The exponential growth model is a mathematical formula used to predict population growth. It assumes that the conditions—such as resources and space—are unlimited, allowing the population to grow continuously at a constant rate. This model is particularly useful in contexts where growth isn't significantly suppressed by external factors.
The key components of the model are:

• \( P(t) \), the population size at a future time
• \( P_0 \), the initial population size
• \( r \), the growth rate as a decimal
• \( t \), the time in years

The formula is expressed as \( P(t) = P_0 (1 + r)^t \). Each variable can impact the overall prediction of future population. By plugging the values into the formula, one can find projections for different time periods. The exponential model is highly beneficial for estimating population trends, but it should be interpreted cautiously in environments where resources or other factors may limit growth realistically.