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?

Short answer

After 1, 2, 10, and 100 years, the population will be 4.644, 4.79, 6.174, and 86.481 million, respectively.

Step-by-step solution

Understand the Growth Formula

The formula for exponential growth is given by: \[ P = P_0 (1 + r)^t \] where \( P \) is the population after time \( t \), \( P_0 \) is the initial population, \( r \) is the growth rate per period, and \( t \) is the number of periods. In this case, \( P_0 = 4.5 \) million, \( r = 0.032 \) (since 3.2% is the same as 0.032), and \( t \) is the number of years.

Calculate the Population After 1 Year

Using the growth formula, substitute \( t = 1 \): \[ P = 4.5 imes (1.032)^1 = 4.5 imes 1.032 \] Calculate \( 4.5 imes 1.032 \) to find the population after 1 year: \[ P = 4.644 \] million. Thus, after 1 year, the population will be 4.644 million.

Calculate the Population After 2 Years

Substitute \( t = 2 \) into the growth formula: \[ P = 4.5 imes (1.032)^2 \] First, calculate \( (1.032)^2 = 1.032 imes 1.032 = 1.065024 \). Then, calculate \( 4.5 imes 1.065024 \) to find the population: \[ P = 4.792608 \] million. Thus, after 2 years, the population will be approximately 4.79 million.

Calculate the Population After 10 Years

Substitute \( t = 10 \) into the growth formula: \[ P = 4.5 imes (1.032)^{10} \] Calculate \( (1.032)^{10} \) using a calculator: \( (1.032)^{10} \approx 1.372 \), then multiply \( 4.5 imes 1.372 \): \[ P = 6.174 \] million. Thus, after 10 years, the population will be approximately 6.174 million.

Calculate the Population After 100 Years

Substitute \( t = 100 \) into the growth formula: \[ P = 4.5 imes (1.032)^{100} \] Calculate \( (1.032)^{100} \) using a calculator (or software, as it is very large): \( (1.032)^{100} \approx 19.218 \), and then multiply \( 4.5 imes 19.218 \): \[ P = 86.481 \] million. So, after 100 years, the population will be approximately 86.481 million.

Key concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population over time. For countries and regions, this is often due to natural factors such as birth rates, death rates, and migration patterns. Understanding how population grows can help us predict future changes and plan for resources, infrastructure, and services.
One characteristic of population growth over time is that it can be exponential. This means that the more people there are, the faster the population can grow, as each individual or pair of individuals can contribute to the production of offspring. Hence, larger populations may increase more rapidly due to the higher number of potential contributors.

Growth Formula

The growth formula is a crucial tool for calculating changes in population over time. In the context of exponential growth, the formula is:

• \( P = P_0 (1 + r)^t \)

where:

• \( P \) is the future population size,
• \( P_0 \) is the initial population size,
• \( r \) is the growth rate per period, expressed as a decimal,
• \( t \) is the number of periods (such as years).

By plugging values into this formula, you can predict the population size after any given period, acknowledging the compound effects of growth over multiple time intervals. For example, a growth rate applied over 10 periods has a larger compounding effect than over a single period.

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of population growth, the function describes how populations multiply over time. This can be represented by the formula:

• \( P = P_0 (1 + r)^t \)

Here, the base \((1 + r)\) is a constant value determined by the growth rate, and \(t\) is the time. The result is that small increases in \(t\) can lead to large increases in \(P\), especially over long periods.
The use of exponential functions allows us to understand compound growth. This is different from linear growth where addition is consistent; exponential growth increases by multiples, which is why populations can increase rapidly when the conditions are right.

Percent Growth Rate

The percent growth rate is a crucial factor in determining how a population will evolve over time. It is expressed as a percentage of the current population size and indicates how much the population increases or decreases within a set period.
For example, a growth rate of 3.2% per year means that each year the population increases by 3.2% of its size at the beginning of that year. To use this rate in calculations, it's often converted to a decimal (for example, 3.2% becomes 0.032) so it can be utilized in the exponential growth formula. The percent growth rate provides insight into how rapidly a population can change, significantly influencing economic planning, resource allocation, and sustainability assessments.