Problem 72 - Contemporary Precalculus

Question

The population of India was approximately 1030 million in 2001 and was 967 million in \(1997 .\) If the population continues to grow exponentially at the same rate, what will it be in \(2010 ?\)

Step-by-Step Solution

Answer

Based on the given data, the estimated population of India in 2010, considering the exponential growth rate, is approximately 1141.48 million.

Step 1: Using the exponential growth formula with given data from 1997 and 2001

We need to set up the exponential growth function for the given years and their corresponding populations. Let's denote the populations of India in 1997 and 2001 as \(P_{1997}\) and \(P_{2001}\), respectively. The exponential growth function for 1997 and 2001 will be: \(P_{1997} = P_{0} e^{r (1997 - 1997)}\) \(P_{2001} = P_{0} e^{r (2001 - 1997)}\) Now, we can substitute the given population values and use these equations to find the growth factor r.

Step 2: Solving for r, the growth rate

We will rewrite the 1997 and 2001 population exponential growth functions using the population values given in the problem. \(967 = P_{0} e^{r (0)}\) \(1030 = P_{0} e^{4r}\) Since \(e^{r(0)} = 1,\) we can rewrite the first equation as: \(P_{0} = 967\) Now, let's plug in the value of \(P_{0}\) in the 2001 equation: \(1030 = 967 e^{4r}\) To find r, we will solve for r in this equation using the following steps: - Divide both sides by 967 \(\frac{1030}{967} = e^{4r}\) - Take the natural logarithm of both sides \(ln(\frac{1030}{967}) = 4r\) - Solve for r \(r = \frac{1}{4} ln(\frac{1030}{967})\)

Step 3: Finding the population in 2010 using the exponential growth formula and the calculated r value

Now that we have the value of r, we can use the exponential growth formula to find the population in 2010: \(P_{2010} = P_{0} e^{r (2010 - 1997)}\) Using the values of \(P_{0}\) and r: \(P_{2010} = 967 e^{\frac{1}{4} ln(\frac{1030}{967}) (2010 - 1997)} \) Calculating the population in 2010: \(P_{2010} \approx 1141.48\) The population of India in 2010 will be approximately 1141.48 million, considering the exponential growth rate estimated from the 1997-2001 population data.

Key Concepts

Population Growth

Population growth refers to the increase in the number of individuals in a population over time. When we talk about population growth in mathematics, we often refer to exponential growth, which is a type of growth where the rate of increase is proportional to the size of the population. This means that as the population becomes larger, it grows more quickly, resembling a curve that rises steeply over time.

Exponential growth is commonly observed in populations that have abundant resources and minimal limitations, such as space or food. However, it's important to note that real-world populations don't grow exponentially forever; they encounter limiting factors that slow down growth eventually.

In our exercise, to estimate the population of India in 2010, we first need to establish the growth pattern from earlier years, using the concept of exponential growth to predict future populations. This method assists governments and organizations in planning for future needs, like food, infrastructure, and healthcare.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. This tool is powerful in predicting future events or behaviors based on current and past data. In our case, we use a mathematical model to describe how the population of India grows over time.

To create an exponential growth model for a population, we typically use the formula:

• \( P(t) = P_0 e^{rt} \)

Where:

• \( P(t) \) is the population at time \( t \)
• \( P_0 \) is the initial population at the starting point
• \( r \) is the growth rate
• \( e \) is the base of the natural logarithm, approximately equal to 2.71828

This formula assumes that the rate of population growth is constant and that it acts continuously, which mirrors the nature of exponential growth in real life. In our step-by-step solution, we determined the constant growth rate \( r \) by analyzing population data from 1997 and 2001. Once we had \( r \), we could then extend our model to predict the population in 2010.

Natural Logarithm

The natural logarithm is a mathematical function denoted by \( \, \ln(x) \, \). It is the inverse of the exponential function and uses the number \( \, e \, \) (approximately 2.71828) as its base. In the context of exponential growth, natural logarithms can help solve equations for the growth rate or other unknowns.

In our exercise, we encounter the natural logarithm when calculating the growth rate \( r \) from the equation:
\[ \ln\left( \frac{1030}{967} \right) = 4r \]
This step involves taking the natural logarithm of both sides of the equation, allowing us to isolate and solve for \( r \). Understanding how natural logarithms work is essential because they provide an insight into how quickly exponential processes grow or decay.

Natural logarithms have applications beyond population growth. They are used extensively in fields like finance, physics, and biology, where exponential changes are common. Recognizing how they transform complex exponential expressions into more manageable linear expressions makes them a valuable tool in mathematics.