Problem 31 - Applied 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 \(2050 .^{\mathrm{s} 1}\) (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

The continuous growth rate is approximately 0.0071. The Day of 7 Billion is predicted to occur around 2012.

Step 1: Understanding Exponential Growth

Exponential growth of a population is described by the formula \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the continuous growth rate, \( t \) is time in years, and \( P(t) \) is the population after time \( t \). We are given \( P_0 = 6.9\) billion in 2010 and \( P(40) = 9 \) billion in 2050 (40 years later). Our goal first is to find \( r \).

Step 2: Setting up the Equation for Growth Rate

Using the exponential growth model formula, we set up the equation with the given values: \[ 9 = 6.9 e^{40r} \]We will solve this equation for \( r \).

Step 3: Solving for the Growth Rate \( r \)

First, divide both sides of the equation by 6.9:\[ e^{40r} = \frac{9}{6.9} \]Now take the natural logarithm (ln) on both sides to solve for \( r \):\[ 40r = \ln\left(\frac{9}{6.9}\right) \]\[ r = \frac{\ln\left(\frac{9}{6.9}\right)}{40} \]Calculate \( r \) using a calculator.

Step 4: Calculating the Year for "Day of 7 Billion"

Use the growth rate \( r \) found in Step 3. The initial population at the end of 2010 was 6.9 billion, and we want to find when it reaches 7 billion:\[ 7 = 6.9 e^{rt} \]\[ e^{rt} = \frac{7}{6.9} \]\[ rt = \ln\left(\frac{7}{6.9}\right) \]\[ t = \frac{\ln\left(\frac{7}{6.9}\right)}{r} \]Solve for \( t \) and add this to the year 2010 to find the predicted year for the population of 7 billion.

Key Concepts

Continuous Growth Rate

When we talk about continuous growth rates, particularly in the context of population growth, we're considering a rate that continuously compounds over time. This means that the growth isn't just happening at the end of each year but is effectively growing at every moment.
This type of growth can be described mathematically using an exponential function where the population at any time, \( P(t) \), depends on its initial population, \( P_0 \), an exponential function of the continuous growth rate, \( r \), and time, \( t \).
The formula used is:

• \( P(t) = P_0 e^{rt} \)

This equation is useful in many real-life scenarios, especially when modeling scenarios like population growth, where the number of individuals is always smoothly changing. If \( r \), the continuous growth rate, is positive, the population will grow; if \( r \) is negative, the population will decline.

Population Modeling

Population modeling via exponential growth is a powerful tool used by scientists and researchers to predict future population sizes.
This kind of modeling helps in understanding the dynamism of a population over time, considering the effect of a constant percentage change that compounds continuously. By inputting specific data points, one can model how a population changes year by year or even decade by decade.
For example, if you're given a starting population and a future population at a specific time, you can apply the exponential growth model to figure out the growth rate. You simply set up an equation and solve for \( r \) using the data points you have, as seen in the original exercise. Once the growth rate is known, further predictions about the population in future years can be made, answering questions such as, "When will the population be 7 billion?" By adjusting the equation, so it equals this new population target, we find the time \( t \), indicating how many years from the starting point this population will be reached.

Natural Logarithm

Natural logarithms, abbreviated as ln, are a key component when dealing with exponential growth equations, particularly when solving for the continuous growth rate \( r \).
The natural logarithm is based on the constant \( e \), which is approximately 2.71828, a number that frequently appears in all kinds of mathematical and scientific applications.
In exponential growth calculations, taking the natural logarithm of both sides of the equation helps to "unpack" the exponent, allowing us to isolate the variables we're interested in, such as solving for \( r \) or \( t \).
For example, in the equation \( e^{40r} = \frac{9}{6.9} \), by applying ln to both sides, you'll simplify it to \( 40r = \ln(\frac{9}{6.9}) \).
This makes natural logarithms a powerful tool for solving equations involving exponential terms. They appear frequently not only in biology and economics but also in physics and engineering when modeling exponential growth or decay phenomena.