Problem 10 - Applied Calculus

Question

World population is approximately \(P=6.4(1.0126)^{t}\) with \(\bar{P}\) in billions and \(t\) in years since 2004 (a) What is the yearly percent rate of growth of the world population? (b) What was the world population in \(2004 ?\) What does this model predict for the world population in \(2010 ?\) (c) Use part (b) to find the average rate of change of the world population between 2004 and 2010

Step-by-Step Solution

Answer

The yearly growth rate is 1.26%. Population in 2004 was 6.4 billion, and in 2010, it was approximately 6.903 billion. The average rate of change is 83.8 million per year.

Step 1: Determine Yearly Growth Rate

The general formula for exponential growth is given by \( P = P_0 (1 + r)^t \), where \( r \) represents the growth rate. In our equation, \( P = 6.4(1.0126)^t \), the growth factor is \( 1.0126 \). Thus, the incremental growth rate \( (r) \) can be determined by noting that it is \( 1.0126 - 1 = 0.0126 \). To express this as a percentage, multiply by 100: \( 0.0126 \times 100 = 1.26\% \).

Step 2: Calculate World Population in 2004

To find the population in 2004, substitute \( t = 0 \) into the equation since 2004 is the starting year. Thus, \( P = 6.4(1.0126)^0 = 6.4 \) billion. Therefore, the population in 2004 was 6.4 billion.

Step 3: Predict World Population in 2010

For the world population in 2010, calculate \( t = 2010 - 2004 = 6 \). Substitute \( t = 6 \) into the equation: \( P = 6.4(1.0126)^6 \). Thus, \( P \approx 6.4 \times 1.0783 \approx 6.9027 \) billion.

Step 4: Average Rate of Change Between 2004 and 2010

The average rate of change is given by the formula \( \frac{P_{2010} - P_{2004}}{2010 - 2004} \). Substitute the values: \( \frac{6.9027 - 6.4}{6} \approx \frac{0.5027}{6} \approx 0.0838 \) billion per year or \( 83.8 \) million per year.

Key Concepts

Rate of Change

The rate of change in the context of population growth tells us how much the population increases or decreases over a specific period of time. When we calculate the average rate of change, we use a specific formula to measure the difference in population between two time points and divide it by the number of years between these points.

For example, when we looked at the world population between 2004 and 2010, we found the population changed from approximately 6.4 billion to about 6.9027 billion. This is a difference of 0.5027 billion. To find the average yearly change, we divided this difference by the number of years, which was 6 years in this case:

• Average rate of change formula: \( \frac{P_{2010} - P_{2004}}{2010 - 2004} \)
• Substituting values: \( \frac{6.9027 - 6.4}{6} \)
• Result: approximately 0.0838 billion or 83.8 million per year

Understanding the average rate of change helps us see the trend and pace of population growth over the period. It's like taking the population growth and spreading it out evenly over several years, giving us a clearer picture of how fast things are changing.

Population Growth

Population growth refers to how the number of individuals in a population increases over time. It's a natural process influenced by birth rates, death rates, immigration, and emigration. In mathematical models, particularly exponential growth models, population growth is often represented using growth factors.

In the example problem, the world population was modeled using the equation \( P = 6.4(1.0126)^t \), where \( P_0 = 6.4 \) represented the approximate population in billions at the start year of 2004, and \( 1.0126 \) was the growth factor.

• The initial population is the starting point, represented by \( P_0 \).
• The growth factor \((1.0126)\) signifies consistent growth over time.
• The exponent \( t \) represents the number of years since the starting year.

Whenever we model population growth exponentially, we assume that resources are unlimited and the growth rate stays constant, which isn't always true in reality. This equation helps predict future populations under these ideal conditions, providing useful insights into how the population may grow over the years.

Yearly Growth Rate

Yearly growth rate is a key concept in understanding how a population is expanding over time. It is especially important in the exponential growth formula where it's represented by the growth factor minus one, then converted into a percentage.

In the given exercise, the growth factor was \( 1.0126 \). This means every year, the population grows by an additional 1.26%. The process to find this growth rate is quite simple:

• Identify the growth factor from the model: \( 1.0126 \)
• Subtract 1 to find the decimal growth rate: \( 1.0126 - 1 = 0.0126 \)
• Convert this decimal to a percentage: \( 0.0126 \times 100 = 1.26\% \)

Understanding the yearly growth rate gives insight into how quickly the population is increasing each year. This annual rate becomes instrumental in planning for resources and infrastructure to support the growing population. It’s a fundamental piece of information for policymakers, scientists, and researchers who are working to predict and manage population growth.