Question

If we were to continue a \(1 \%\) population growth trajectory into the future, work out how many years it would take to go from 7 billion people to 8 billion, and then from 8 billion to 9 billion.

Short answer

Answer: It takes approximately 13.21 years for the population to grow from 7 billion to 8 billion and approximately 11.82 years to grow from 8 billion to 9 billion with a 1% growth rate.

Step-by-step solution

Determine the growth rate

The growth rate (\(r\)) is given as 1%, so we need to convert this percentage into a decimal. To do that, divide the percentage by 100: \(r = \frac{1}{100} = 0.01\).

7 billion to 8 billion

Now let's use the exponential growth formula to find the number of years (\(t\)) it takes to grow the population from 7 billion to 8 billion. \(8,000,000,000 = 7,000,000,000(1 + 0.01)^t\) First, we need to isolate the variable (\(t\)): \(\frac{8,000,000,000}{7,000,000,000} = (1 + 0.01)^t\) Now, calculate the value of the left side of the equation: \(\approx 1.142857 \) And now, rewrite the equation: \(1.142857 = (1.01)^t\) In order to solve this equation for \(t\), we use the logarithm property \(\log_b(a^x) = x\log_b(a)\). Taking the natural logarithm (log base e) of both sides: \(\ln(1.142857) = \ln((1.01)^t)\) Applying the logarithm property: \(t\ln(1.01) = \ln(1.142857)\) Finally, solve for \(t\) by dividing both sides by \(\ln(1.01)\): \(t = \frac{\ln(1.142857)}{\ln(1.01)}\) Calculate the result: \(t \approx 13.21\) It takes approximately 13.21 years for the population to grow from 7 billion to 8 billion.

8 billion to 9 billion

Now let's use the exponential growth formula to find the number of years (\(t\)) it takes to grow the population from 8 billion to 9 billion. \(9,000,000,000 = 8,000,000,000(1 + 0.01)^t\) Again, isolate the variable (\(t\)): \(\frac{9,000,000,000}{8,000,000,000} = (1 + 0.01)^t\) Calculate the left side of the equation: \(\approx 1.125\) Rewrite the equation: \(1.125 = (1.01)^t\) Taking the natural logarithm of both sides: \(\ln(1.125) = \ln((1.01)^t)\) Applying the logarithm property: \(t\ln(1.01) = \ln(1.125)\) Finally, solve for \(t\) by dividing both sides by \(\ln(1.01)\): \(t = \frac{\ln(1.125)}{\ln(1.01)}\) Calculate the result: \(t \approx 11.82\) It takes approximately 11.82 years for the population to grow from 8 billion to 9 billion. In conclusion, it takes approximately 13.21 years for the population to grow from 7 billion to 8 billion and approximately 11.82 years to grow from 8 billion to 9 billion with a 1% growth rate.

Key concepts

Exponential Growth Formula

Population growth can be predicted through the exponential growth formula. This formula is useful when dealing with consistent percentage growth rates, as it helps estimate how long it will take for a population to reach a certain size. The exponential growth formula is given by the equation: \[ P(t) = P_0(1 + r)^t \] where:

• \( P(t) \) is the future population size.
• \( P_0 \) is the initial population size.
• \( r \) is the growth rate expressed as a decimal.
• \( t \) is the time in years.

To solve problems using this formula, the objective is typically to find \( t \), the number of years required for the population to reach the future size \( P(t) \). The process involves rearranging the formula and often using logarithms to solve for \( t \). This formula is powerful in modeling population dynamics over time, especially when growth remains constant.

Natural Logarithm

The natural logarithm is an essential tool when dealing with exponential growth problems. Denoted as \( \ln \), it is the logarithm to the base of the mathematical constant \( e \), approximately equal to 2.71828. The natural logarithm simplifies calculations of growth over time, especially when isolating the time variable in the exponential formula. To solve for time \( t \) when the formula \( P(t) = P_0(1 + r)^t \) is involved, we rewrite it as \((1 + r)^t = \frac{P(t)}{P_0}\) and take the natural logarithm of both sides: \[ \ln((1 + r)^t) = \ln\left(\frac{P(t)}{P_0}\right) \] Utilizing the power rule of logarithms, where \( \log_b(a^c) = c \cdot \log_b(a) \), the equation becomes: \[ t \cdot \ln(1 + r) = \ln\left(\frac{P(t)}{P_0}\right) \] Solving for \( t \) involves dividing both sides of the equation by \( \ln(1 + r) \), which provides a clear path to finding the time required for population growth.

Growth Rate Interpretation

Understanding the growth rate is crucial in population growth calculations. The growth rate reflects how quickly a population increases over a given period, typically expressed as a percentage. When using the exponential growth formula, you need to convert this percentage into a decimal by dividing it by 100. For example, a 1% growth rate becomes \( 0.01 \). In exponential growth calculations, the growth rate determines how fast the initial population size will multiply each year. A small difference in growth rates can significantly impact the time taken to reach a target population as the effect compounds over time. This is why accurately interpreting and applying the growth rate is essential for precise predictions. For instance, in population growth problems, distinguishing between whether a 1% growth rate is optimistic or pessimistic can influence potential planning and policy decisions. Recognizing how this rate affects exponential growth allows for better strategic forecasts and effective resource allocation.