Question

Statistics indicate that the world population since 1995 has been growing at a rate of about \(1.27 \%\) per year. United Nations records estimate that the world population in 2011 was approximately 7 billion. Assuming the same exponential growth rate, when will the population of the world be 9 billion?

Short answer

The world population will be 9 billion by the year 2031.

Step-by-step solution

Identify the formula

The population growth can be modeled using the exponential growth formula: \( P(t) = P_0 \times e^{rt} \) where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years.

Substitute the known values

Given: Initial population (\(P_0\)) = 7 billion, Growth rate (\(r\)) = 1.27% = 0.0127, Future population (\(P(t)\)) = 9 billion. Substitute these values into the formula: \( 9 = 7 \times e^{0.0127t} \)

Solve for \(t\)

First, divide both sides by 7: \( \frac{9}{7} = e^{0.0127t} \). Then take the natural logarithm of both sides: \( \text{ln}(\frac{9}{7}) = 0.0127t \). Now, solve for \( t \) by dividing by 0.0127: \( t = \frac{\text{ln}(\frac{9}{7})}{0.0127} \).

Calculate the value

Using a calculator to find the natural logarithm: \( \text{ln}(\frac{9}{7}) \approx 0.2513 \). Then, divide by the growth rate: \( t \approx \frac{0.2513}{0.0127} \approx 19.78 \). This means approximately 20 years from 2011.

Find the year

Add the number of years to 2011: \(2011 + 20 = 2031 \). Thus, the world population is expected to reach 9 billion by the year 2031.

Key concepts

Exponential Growth Formula

Exponential population growth is modeled using a mathematical formula that helps us predict future population sizes. The exponential growth formula is:
\[ P(t) = P_0 \times e^{rt} \] Here,

• \( P(t) \) represents the future population.
• \( P_0 \) is the initial or starting population.
• \( r \) is the growth rate.
• \( t \) is the time in years.

Essential components of this formula, like the base of the natural logarithm \( e \), ensure the model accurately captures the continuous growth trend that real-world populations often exhibit. By filling in the known values, we can solve for the time it will take for a population to grow to a certain size. This simple, yet highly informative formula provides a clear pathway to understand population dynamics.

Natural Logarithm

The natural logarithm (often abbreviated as 'ln') is a fundamental concept when working with exponential growth. It is the inverse operation of the exponential function. In our formula, we often need to apply the natural logarithm to isolate the variable of interest, like time \( t \). For example, to solve for \( t \) in the equation \[ e^{0.0127t} = \frac{9}{7} \] we apply the natural logarithm to both sides: \[ \text{ln} \left(\frac{9}{7}\right) = 0.0127t \] This step transforms the exponential equation into a linear one, making it much simpler to solve for \( t \). The natural logarithm is particularly useful because it directly relates to the continuous growth rate using base \( e \), which is approximately 2.718.

Population Growth Rate

The population growth rate is a critical factor in the exponential growth model. In our example, the growth rate \( r \) is given as 1.27%, or 0.0127 in decimal form. This rate is an annual increase in the population size, expressed as a percentage of the current population. To use this growth rate in our calculations, we convert the percentage to a decimal by dividing by 100. When substituted into our exponential growth formula, it determines how rapidly the population grows over time. For instance, a growth rate of 1.27% means that every year, the population increases by 1.27% of its size from the previous year, compounded continuously.

Mathematical Modeling

Mathematical modeling is a powerful tool used to represent real-world phenomena through mathematical equations. In the context of population growth, it allows us to predict future population sizes based on current data and assumed growth rates. The exponential growth model \( P(t) = P_0 \times e^{rt} \) is a type of mathematical model used to describe situations where growth happens continually and proportionally to the current size. By inputting known values and solving for the unknown variable, we make informed predictions about future events. This modeling helps in planning and policy-making by providing a clear picture of potential future scenarios.