Problem 9 - Calculus Early Transcendentals: Pearson New International Edition

Question

The population of a certain country is growing at \(3.2 \%\) per year; that is, if it is \(A\) at the beginning of a year, it is \(1.032 \mathrm{~A}\) at the end of that year. Assuming that it is \(4.5\) million now, what will it be at the end of 1 year? 2 years? 10 years? 100 years?

Step-by-Step Solution

Answer

After 1 year: 4.644 million; 2 years: 4.793 million; 10 years: 6.186 million; 100 years: 862.385 million.

Step 1: Understand the Growth Formula

The problem states that the population grows by \(3.2\%\) per year. If the current population is \(A\), at the end of one year, it will be \(1.032 \times A\). This is an example of exponential growth, which follows the formula: \(P_n = P_0 \times (1+r)^n\), where \(P_0\) is the initial population, \(r\) is the growth rate, and \(n\) is the number of years.

Step 2: Identify the Initial Population and Growth Rate

The initial population \(P_0\) is \(4.5\) million and the growth rate \(r\) is \(0.032\) (since 3.2% growth translates to 0.032 in decimal form). We will use these values to compute future populations.

Step 3: Calculate Population After 1 Year

Using the formula \(P_1 = P_0 \times (1+r)\), calculate the population at the end of 1 year. Substitute \(P_0 = 4.5\) million and \(r = 0.032\): \[P_1 = 4.5 \times 1.032 = 4.644\text{ million}\]

Step 4: Calculate Population After 2 Years

To find the population after 2 years, use the formula \(P_2 = P_0 \times (1+r)^2\). Substitute the known values:\[P_2 = 4.5 \times (1.032)^2 = 4.5 \times 1.065024 = 4.792608\text{ million}\]

Step 5: Calculate Population After 10 Years

Now calculate the population after 10 years with \(P_{10} = P_0 \times (1+r)^{10}\):\[P_{10} = 4.5 \times (1.032)^{10} = 4.5 \times 1.3746964 = 6.1861338\text{ million}\]

Step 6: Calculate Population After 100 Years

For the population after 100 years, use \(P_{100} = P_0 \times (1+r)^{100}\):\[P_{100} = 4.5 \times (1.032)^{100} \approx 4.5 \times 191.641 = 862.3845\text{ million}\]

Step 7: Conclusion

The populations at the end of 1, 2, 10, and 100 years are 4.644 million, 4.793 million, 6.186 million, and approximately 862.385 million, respectively.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population. Typically, this growth occurs due to natural mechanisms such as births or migration in and out of an area. In this example, we consider the situation where a country's population grows steadily by 3.2% each year.

When examining population growth, it's essential to note that:

• The population growth rate can be constant or variable.
• A constant growth rate, as in this scenario, leads to exponential growth.
• Factors such as resources, birth rates, and mortality rates often impact real-world population growth.

Understanding the dynamics of population growth is vital for planning and decision-making in areas such as resource allocation and infrastructure development.

Mathematical Modeling

Mathematical modeling is a powerful tool that helps us understand and predict real-life phenomena by using mathematical expressions. In the context of population growth, mathematical modeling allows us to predict future population sizes based on current data and assumed conditions.

The model applied here uses the exponential growth formula, which is expressed as:\[P_n = P_0 \times (1+r)^n\]where:

• \(P_0\) is the initial population size.
• \(r\) is the growth rate as a decimal.
• \(n\) represents the number of years into the future.

By substituting known values into this equation, we can effectively model how the population changes over time. Mathematical models like this are invaluable because they provide quantitative insights that are crucial for policy-making and strategic planning.

Exponential Functions

Exponential functions describe situations where quantities change at rates proportional to their current value. In population growth, this means the population increases a fixed percentage each year, resulting in exponential growth.

Key characteristics of exponential functions include:

• An initial value that changes by a consistent percentage over time.
• A growth factor (base) greater than 1, which in this case is 1.032 (3.2% growth).
• Rapid increases, especially noticeable over longer time periods.

Exponential functions are expressed generally as:\[y = a \times b^x\]where \(a\) is the initial value, \(b\) is the growth factor, and \(x\) is the time period. Recognizing the exponential nature of growth helps in understanding how even modest growth rates can lead to significant increases over time in a population's size.