Problem 28 - Applied Calculus

Question

In \(2012,\) the world's population was 7 billion, and the population is projected \(^{79}\) to reach 8 billion by the year 2025. What annual growth rate is projected?

Step-by-Step Solution

Answer

The projected annual growth rate is approximately 1.06%.

Step 1: Understand the Problem

We need to find the annual growth rate, which means determining the constant percentage by which the population grows each year from 2012 to 2025. The population grows from 7 billion to 8 billion during this period.

Step 2: Use the Exponential Growth Formula

The formula for exponential growth is given by \( P = P_0(1 + r)^t \), where \( P \) is the final population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the number of years. In this problem, \( P = 8 \) billion, \( P_0 = 7 \) billion, and \( t = 2025 - 2012 = 13 \) years.

Step 3: Substitute Values into the Formula

Substitute the given values into the formula: \[ 8 = 7(1 + r)^{13} \].

Step 4: Solve for the Growth Rate

Divide both sides of the equation by 7 to isolate \((1 + r)^{13}\): \[ \frac{8}{7} = (1 + r)^{13} \].

Step 5: Calculate 1 + r

Find \((1 + r)\) by taking the 13th root of both sides: \[ 1 + r = \left(\frac{8}{7}\right)^{\frac{1}{13}} \].

Step 6: Determine the Growth Rate \(r\)

Subtract 1 from both sides to find \(r\): \[ r = \left(\frac{8}{7}\right)^{\frac{1}{13}} - 1 \].

Step 7: Calculate Using a Calculator

Use a calculator to evaluate \( r \). After computing, you find that \( r \approx 0.0106 \), which is equivalent to 1.06% annual growth rate.

Key Concepts

Annual Growth Rate

The annual growth rate is a crucial measure to understand how quickly a population increases over time. In essence, it's a percentage that shows us by how much the population size has increased in a year. When dealing with exponential growth, as in our exercise, the population doesn't grow by the same fixed amount every year. Instead, the growth is compounded, meaning each year's growth builds on the previous year’s population. Calculating the annual growth rate involves:

• Identifying the initial population (in this case, the world's population in 2012, which was 7 billion).
• Finding the final projected population after a set period (8 billion by 2025).
• Determining the number of years over which this growth occurs (13 years from 2012 to 2025).

In our example, the growth rate was calculated to be approximately 1.06%. This is derived from using the exponential growth formula, ensuring you consider the compound nature of growth.

Population Projection

Population projection helps us forecast future population sizes, accounting for variables like birth rates, death rates, and migration patterns. This is invaluable for planning in areas like resource allocation, urban planning, and environmental conservation. To make a population projection, you typically need:

• An initial population size at a known starting point.
• Projected growth rates (like our calculated 1.06% annual growth).
• The time span over which the projection is made.

These projections use exponential growth principles, recognizing that populations tend to grow at rates proportional to their current size. This exercise demonstrates projecting population size from 7 billion in 2012 to 8 billion by 2025, thanks to a consistent annual growth rate.

Exponential Growth Formula

The exponential growth formula is pivotal for calculating changes in populations where growth occurs at a constant percentage rate. This mathematical expression is written as: \[ P = P_0(1 + r)^t \]Where:

• \( P \) is the future population.
• \( P_0 \) represents the initial population size.
• \( r \) is the annual growth rate (expressed as a decimal).
• \( t \) is the time duration in years.

In the exercise provided, solving for \( r \), involves manipulating this formula to find our unknown, the annual growth rate.Here's the process:1. Insert the known values into the equation: \[ 8 = 7(1 + r)^{13} \]2. Solve for the growth factor \((1 + r)\) by dividing both sides by the initial population (7 billion).3. Use logarithms or a calculator to find \( r \), resulting in the annual growth rate.This formula is crucial for many real-world applications, enabling us to predict future population sizes accurately.