Problem 24 - Applied Calculus

Question

If the world's population increased exponentially from 5.937 billion in 1998 to 6.771 billion in 2008 and continued to increase at the same percentage rate between 2008 and \(2012,\) calculate what the world's population would have been in \(2012 .\) How does this compare to the Population Reference Bureau estimate of 7.07 billion in July \(2012 ?^{5 x}\).

Step-by-Step Solution

Answer

The population in 2012 was about 7.148 billion, slightly higher than the estimate of 7.07 billion.

Step 1: Calculating the Growth Rate

To determine the growth rate, we need to find the rate at which the population increased from 1998 to 2008. We use the formula for exponential growth: \[ P = P_0 \times e^{rt} \]where:- \( P \) is the final population size (6.771 billion),- \( P_0 \) is the initial population size (5.937 billion),- \( r \) is the growth rate,- \( t \) is the time in years (10 years),- \( e \) is the base of the natural logarithm.Rearranging the formula to solve for \( r \), we have:\[ r = \frac{\ln(\frac{P}{P_0})}{t} \]Substitute the given values to find \( r \):\[ r = \frac{\ln(\frac{6.771}{5.937})}{10} \approx 0.013 \text{ or } 1.3\% \text{ per year} \]

Step 2: Projecting Population to 2012

Now that we have the annual growth rate of 1.3%, use the exponential growth formula to calculate the population in 2012.We know:- 2008 population \( P_0 = 6.771 \) billion,- \( r = 0.013 \),- \( t = 4 \) years (from 2008 to 2012).Substitute these into the formula:\[ P = 6.771 \times e^{0.013 \times 4} \approx 7.148 \text{ billion} \]

Step 3: Comparison with Estimate

The calculated population in 2012 is approximately 7.148 billion. The Population Reference Bureau estimated the population to be 7.07 billion in July 2012. Comparing these: 7.148 billion (calculated) is slightly higher than 7.07 billion (estimated). The difference is approximately 0.078 billion or 78 million.

Key Concepts

Exponential Growth

Exponential growth is a fascinating concept in mathematics and real-world applications. It occurs when the rate of growth of a population or any other quantity is proportional to its current size. This means that the larger the population gets, the faster it grows. In mathematical terms, we present exponential growth with the formula: \[ P = P_0 \times e^{rt} \]where:

• \( P \) represents the final population size.
• \( P_0 \) is the initial population size.
• \( r \) stands for the growth rate.
• \( t \) is the time over which the growth occurs.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.

Exponential growth is common in biological settings when resources are abundant. This is because as more individuals are born, there are more entities to reproduce, leading to faster population increases.

Growth Rate Calculation

Determining the growth rate is crucial for understanding how quickly a population grows over time. To calculate this rate for exponential growth, we often rearrange our initial formula. The rearranged formula used to find the growth rate \( r \) is:\[ r = \frac{\ln(\frac{P}{P_0})}{t} \]Let's break down the formula's components:

• \( \ln \) represents the natural logarithm, which helps linearize exponential data.
• \( \frac{P}{P_0} \) is the ratio of the final population to the initial population.
• \( t \) signifies the period over which the growth occurs.

By dividing the logarithm of the ratio by the time span, we can determine the annual percentage by which the population is increasing. In our example, the growth rate was approximately 1.3% per year, showing a consistent annual increase over the evaluated period.

World Population Estimation

World population estimation involves forecasting future population sizes based on current trends and historical data. It is often utilized by governments and organizations to plan for future needs, such as food, housing, and infrastructure. Using the growth rate, we calculate projected populations by applying the exponential growth formula again. In our exercise, we estimated the population in 2012 by continuing the growth trend observed from 1998 to 2008. By inputting the 2008 population (\(6.771\) billion), the growth rate (\(0.013\) or 1.3%), and a time period of 4 years, we reached an estimated population of approximately 7.148 billion. It's important to compare calculated estimates to actual data, like the Population Reference Bureau's estimate for 2012, which was 7.07 billion. Such comparisons help validate our calculations or might indicate variations due to unforeseen demographic or socio-economic factors. An understanding of these estimates and their comparisons can be crucial for effective planning and policy making.