Problem 47 - Applied Calculus

Question

In \(2011,\) the populations of China and India were approximately 1.34 and 1.19 billion people \(^{69},\) respectively. However, due to central control the annual population growth rate of China was \(0.4 \%\) while the population of India was growing by \(1.37 \%\) each year. If these growth rates remain constant, when will the population of India exceed that of China?

Step-by-Step Solution

Answer

The population of India will exceed that of China in 2043.

Step 1: Understand the Problem

We need to determine the year when the population of India will surpass that of China given their respective growth rates. China starts with 1.34 billion people growing at 0.4% annually, and India starts with 1.19 billion people growing at 1.37% annually.

Step 2: Set Up Exponential Growth Models

The population growth for both countries can be described by exponential functions. Let \( P_c(t) \) be the population of China and \( P_i(t) \) the population of India at year \( t \), where year 0 is 2011. So, \( P_c(t) = 1.34 \times (1 + 0.004)^t \) and \( P_i(t) = 1.19 \times (1 + 0.0137)^t \).

Step 3: Establish the Inequality for Population Exceedance

We want to find the smallest year \( t \) such that \( P_i(t) > P_c(t) \). This leads to the inequality: \[ 1.19 \times (1.0137)^t > 1.34 \times (1.004)^t \].

Step 4: Solve the Inequality Algebraically

Divide both sides of the inequality by \( 1.19 \) and \( (1.004)^t \), giving \( (1.0137/1.004)^t > 1.34/1.19 \). Simplify this to \( (1.0095118)^t > 1.12605 \).

Step 5: Take the Natural Logarithm

Taking the natural logarithm of both sides, we have \( t \cdot \ln(1.0095118) > \ln(1.12605) \).

Step 6: Solve for t

Divide to isolate \( t \): \[ t > \frac{\ln(1.12605)}{\ln(1.0095118)} \]. Calculating this gives \( t \approx 32.02 \).

Step 7: Determine the Year

Since \( t \approx 32.02 \), India will surpass China in population approximately 32 years after 2011, which is the year 2043.

Key Concepts

Exponential Growth

Exponential growth is a process where something increases by a constant percentage over equal time periods. Many natural phenomena, including population growth, follow this pattern. In our problem, both China and India's populations are growing exponentially, but at different rates. To describe exponential growth mathematically, we use functions. For China's population, you start with 1.34 billion and multiply it by an annual growth factor of 1.004. Similarly for India, the starting population is 1.19 billion, and the annual growth factor is 1.0137.
These functions allow us to predict how populations will grow in the future. They look like this:

• China: \( P_c(t) = 1.34 \times (1.004)^t \)
• India: \( P_i(t) = 1.19 \times (1.0137)^t \)

The t in these equations represents the number of years after 2011. Thus, understanding exponential growth is key to predicting when India's population will eventually surpass that of China.

Inequality Solving

To find out when India's population will exceed China's, we used inequality solving. An inequality is a mathematical statement that shows one expression is greater than or less than another. In this case, we're interested in when India's population expression becomes greater than China's. This is expressed with the inequality:
\[ 1.19 \times (1.0137)^t > 1.34 \times (1.004)^t \]
The process involves manipulating both sides of this inequality to isolate the growth factors on one side. By simplifying, dividing, and rearranging terms, we end up with:
\[ (1.0137/1.004)^t > 1.34/1.19 \]
This form of an inequality lets us directly compare the relative growth speeds of the two populations, crucial for determining when India overtakes China.

Algebraic Manipulation

Algebraic manipulation involves changing the structure of an equation or inequality to make it easier to solve. In this exercise, it was necessary to break down the initial inequality to a form where we could isolate the variable t. This was done by dividing both sides by certain constants and rearranging equations.
Next, we utilized logarithms, which are inverse operations to exponents, to handle the exponential parts. Taking natural logarithms on both sides transforms the exponential inequality into a simple linear one:
\[ t \cdot \ln(1.0095118) > \ln(1.12605) \]
The process involves isolating t by dividing both sides by \( \ln(1.0095118) \). This manipulation allows us to solve for t directly, leading to:
\[ t > \frac{\ln(1.12605)}{\ln(1.0095118)} \]
A key takeaway is that algebraic manipulation is crucial when dealing with exponential inequalities, as regular algebraic methods won't suffice.