Problem 47 - Calculus

Question

The world population was 6.9 billion at the end of 2010 and is predicted to reach 9 billion by the end of \(2050 .^{32}\) (a) Assuming the population is growing exponentially, what is the continuous growth rate per year? (b) The United Nations celebrated the "Day of 5 Billion" on July \(11,1987,\) and the "Day of 6 Billion" on October \(12,1999 .\) Using the growth rate in part (a), when is the "Day of 7 Billion" predicted to be?

Step-by-Step Solution

Answer

(a) The continuous growth rate is 0.6873% per year. (b) "Day of 7 Billion" is predicted around February 2014.

Step 1: Understand Exponential Growth Formula

The formula for exponential growth is given by: \[ P(t) = P_0 \cdot e^{rt} \] where \( P(t) \) is the future population, \( P_0 \) is the initial population, \( r \) is the continuous growth rate, and \( t \) is the time in years.

Step 2: Solve for Growth Rate

Using the exponential growth formula, set up the equation for the population growth from 2010 to 2050:\[ 9 = 6.9 \cdot e^{r \cdot 40} \]Solve for \( r \):1. Divide both sides by 6.9:\[ e^{40r} = \frac{9}{6.9} \]2. Take the natural logarithm of both sides:\[ 40r = \ln\left(\frac{9}{6.9}\right) \]3. Solve for \( r \):\[ r = \frac{\ln\left(\frac{9}{6.9}\right)}{40} \]Calculate to find the continuous growth rate \( r \approx 0.006873 \) or \( 0.6873\% \) per year.

Step 3: Use Growth Rate to Predict Day of 7 Billion

Estimate when the population would reach 7 billion using the growth rate found:1. Set up the equation: \[ 7 = 6 \cdot e^{0.006873 \cdot t} \] since the last known population was 6 billion in 1999.2. Solve for \( t \):\[ e^{0.006873 \cdot t} = \frac{7}{6} \]3. Take the logarithm of both sides:\[ 0.006873 \cdot t = \ln\left(\frac{7}{6}\right) \]4. Solve for \( t \):\[ t = \frac{\ln\left(\frac{7}{6}\right)}{0.006873} \]5. Calculate \( t \) to find the years after 1999. \( t \approx 14.834 \) years.Therefore, the population reaches 7 billion around February 2014.

Key Concepts

Continuous Growth Rate

In the context of population growth, the continuous growth rate is a measure of how fast the population is increasing each year, assuming the growth follows a smooth exponential pattern. This rate is a constant that you multiply over time to predict future values of population from an initial value. Think of it as a percentage increase that continuously compounds.

Understanding and determining the continuous growth rate involves using a specific formula, which we'll get to in detail later. For now, it's essential to understand that this rate is crucial for accurate long-term predictions in fields like biology, demography, and economics. A higher continuous growth rate indicates a faster expansion of the population, while a lower rate suggests a slower growth. Let's dive into how you can calculate this rate using the natural logarithm.

Population Prediction

Predicting population size over time helps governments and organizations plan for the future. To predict how the population will change, one often uses mathematical models like exponential growth, which captures patterns seen in natural world growth scenarios.

The key to successful population prediction lies in understanding past trends and utilizing known mathematical principles. When we apply the continuous growth rate, we can estimate population at any future time given an initial population. For instance, using past data, we can estimate when global population milestones, such as the Day of 7 Billion, will happen. This involves setting up an exponential growth equation with known variables to solve for *t*, which gives the number of years into the future when the population hits a certain number.

Natural Logarithm

The natural logarithm, commonly denoted as \( \ln \), is a mathematical function that is pivotal in solving exponential equations—especially those related to continuous growth. It is the inverse operation of taking \( e \) to a power, where \( e \) is the base of the natural logarithms and approximately equals 2.718.

In practical terms, when you have an equation with an exponential term that contains a variable in its exponent, taking the natural logarithm allows you to solve for that variable. For population predictions, once you set up your exponential growth equation, using \( \ln \) helps isolate the variable representing time or the growth rate. This is done by taking the natural logarithm of both sides of the equation, turning a problem of exponentiation into a more manageable multiplication and division exercise.

Exponential Growth Formula

The exponential growth formula is a mathematical representation used to model situations where growth accelerates over time. This formula, \( P(t) = P_0 \cdot e^{rt} \), expresses how a quantity grows exponentially, where \( P_0 \) is the initial amount, \( r \) is the growth rate, and \( t \) is time.

To understand its application, consider how populations grow. Populations tend not to grow linearly due to the compounding nature of reproduction. Instead, they follow an exponential curve where each increase builds upon the last. Using this formula, you can predict future population given current data, reflecting patterns not just of sheer numbers but also of rate dynamics. It allows us to see the effect of continuous compounding—in population, finance, or any field where this type of growth is relevant.