Problem 24 - Calculus Early Transcendentals

Question

Devise an exponential decay function that fits the following data; then answer the accompanying questions. Be sure to identify the reference point \((t=0)\) and unitsof time. China's one-child policy was implemented with a goal of reducing China's population to 700 million by 2050 (from 1.2 billion in 2000). Suppose China's population declines at a rate of \(0.5 \% /\) yr. Will this rate of decline be sufficient to meet the goal?

Step-by-Step Solution

Answer

Answer: No, the one-child policy will not be sufficient to achieve the population goal of 700 million by 2050 at the given rate of decline.

Step 1: Define the exponential decay function

The general form of an exponential decay function is \(P(t) = P_0 e^{-kt}\), where \(P(t)\) is the population at time \(t\), \(P_0\) is the initial population, \(k\) is the decay rate, and \(t\) is the time in years. Given initial population, \(P_0 = 1.2 \times 10^9\), and decay rate per year, \(k = 0.5\% = 0.005\). The exponential decay function is \(P(t) = (1.2 \times 10^9) e^{-0.005t}\).

Step 2: Identify the reference point and the units of time

The reference point is when \(t=0\). In the problem, we are given the initial population at the year 2000. So, the reference point is the year 2000 \((t=0)\). The units of time are in years as we are measuring the population decline annually.

Step 3: Calculate the population in 2050 using the exponential decay function

To find the population of China in 2050, we need to calculate \(P(2050-2000) = P(50)\). Using the decay function, \(P(t) = (1.2 \times 10^9) e^{-0.005t}\), we find the population in the year 2050: \(P(50) = (1.2 \times 10^9) e^{-0.005(50)} \approx 8.29 \times 10^8\).

Step 4: Compare the obtained population value with the target population

Now, let's see if the population goal of 700 million, i.e., \(7 \times 10^8\), will be achieved by 2050. The population value we got from the exponential decay function is \(8.29 \times 10^8\), which is greater than the target population of \(7 \times 10^8\). So, at the given rate of decline, the one-child policy will not be sufficient to reduce the population to the desired goal of 700 million by 2050.

Key Concepts

Population Decline

Population decline refers to the reduction in a population size over time. This can occur for several reasons, including low birth rates, high death rates, and emigration. In the context of the one-child policy, it was a government mandate designed to slow the rapid population growth in China.
Implementing such policies often leads to a phenomenon called exponential decay, where the population decreases at a rate proportional to its current size.
This can result in various social and economic consequences, such as:

• Labor shortages due to fewer young people entering the workforce.
• Increased elder care needs as the proportion of older individuals rises.
• Changes in economic growth patterns as consumer demand shifts.

Understanding these dynamics is crucial for planning and implementing policies that manage population changes effectively.

One-Child Policy

The one-child policy of China was a government-initiated measure launched in 1980 to control population growth. The policy limited urban couples to having only one child, although exceptions were made for certain groups, like ethnic minorities.
The aim was to curb the population explosion and conserve resources for sustainable economic development. While the policy decreased the birth rate, it also led to several unintended consequences:

• Gender imbalance due to a cultural preference for male children.
• Aging population and a shrinking workforce as fewer young people were born.
• Increased pressure on the single child to support aging parents.

The policy was relaxed to a two-child policy in 2016, reflecting the challenges posed by rapid population aging and the need for a balanced population structure.

Mathematical Model

Mathematical models use mathematical structures and concepts to study real-world systems. They help us predict outcomes based on certain inputs.
An exponential decay model is one of these mathematical tools, often used to model population decline, radioactive decay, and cooling coffee.
In the exponential decay equation, \(P(t) = P_0 e^{-kt}\):

• \(P(t)\) represents the population at time \(t\).
• \(P_0\) is the initial population.
• \(k\) is the decay rate.
• \(t\) is time.

This model assumes that each year's population declines by the same percentage, making it a useful tool for predicting future population sizes based on current data. It shows us that China's population, starting at 1.2 billion with a 0.5% decrease per year, would not reach the target of 700 million by 2050 under the current rate of decline without additional measures.