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 units of 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, based on the exponential decay function, China's population is estimated to be approximately 774 million in 2050, which is not sufficient to meet the population goal of 700 million by 2050.

Step 1: 1. Define the exponential decay function

An exponential decay function can be written in the form \(P(t) = P_0e^{-kt}\), where \(P(t)\) represents the population at time \(t\), \(P_0\) is the initial population, \(k\) is the decay rate as a positive constant, and \(t\) is the time in years.

Step 2: 2. Identify the initial population and decay rate

We are given that the initial population (\(P_0\)) is 1.2 billion in 2000, and the decay rate is 0.5% per year. We need to convert the percentage decay rate into a decimal by dividing by 100. So, \(k = 0.5/100 = 0.005\).

Step 3: 3. Plug in the initial population and decay rate into the decay function

Now that we have the initial population and decay rate, we can plug them into the exponential decay function: \(P(t) = 1.2 \times 10^9e^{-0.005t}\).

Step 4: 4. Identify the reference point for time

The reference point is when \(t=0\). Since the population data is given for the year 2000, we can say that our reference point is \((t=0) = 2000\) and the units of time are years.

Step 5: 5. Calculate the population in 2050 using the decay function

We are interested in finding the population in 2050, which is 50 years after the reference year 2000. So, we need to plug in \(t=50\) into our decay function: \(P(50) = 1.2 \times 10^9e^{-0.005 \times 50}\). Computing this, we get: \(P(50) \approx 774,158,703\).

Step 6: 6. Determine if the population goal is met by 2050

The goal is to reach a population of 700 million by 2050. From our calculations in step 5, we found that the population would be approximately 774 million in 2050. Therefore, the decay rate of 0.5% per year is not sufficient to meet the population goal of 700 million by 2050.

Key Concepts

Population Dynamics

Population dynamics is all about the changes in population size and composition over time. It's influenced by various factors such as birth rates, death rates, immigration, and emigration. For understanding how a population grows or declines, knowing these dynamics is crucial.

When we talk about exponential decay in population dynamics, we focus on how populations decrease over time due to set factors like a decrease in birth rates or policies like China's one-child policy. These factors influence the rate at which a population shrinks or grows.

In the exercise, the central role of population dynamics is seen through China's strategic attempt to control its population through a defined policy. The goal was to bring down the population to a specific number by a target year, which clearly demonstrates managing population sizes strategically based on policy-guided population dynamics.

Understanding these dynamics provides insights into how well we can predict population changes and implement policies effectively to control future outcomes.

Decay Rate Calculation

Decay rate calculation is foundational in modeling how quickly or slowly a quantity decreases over time. In exponential decay, like in our exercise, this rate is crucial for predicting future values accurately.

The decay rate, represented as a constant 'k' in the exponential decay function, shows how fast the initial amount diminishes. In the equation, it's crucial to have the decay rate as a positive constant that modulates the exponential function:

• The decay rate is provided as a percentage.
• To use it in equations, convert this percentage into a decimal. For instance, 0.5% becomes 0.005. This conversion is essential to properly compute the exponential decay function.

Plugging this decay rate into our formula \[ P(t) = P_0e^{-kt} \]helps us calculate future population sizes. In our example, expressing 0.5% as 0.005 allowed us to solve for the projected population in the year 2050, highlighting the straightforward but critical nature of decay rate calculation in such projections.

Exponential Functions

Exponential functions, portrayed as \( P(t) = P_0e^{-kt} \), are pivotal for understanding how values grow or decline rapidly over time. This formula is central to both exponential growth and decay models.

For decay scenarios, like reducing a population, the function includes the initial amount, the constant decay rate, and time as variables. Here's a quick breakdown:

• \( P_0 \) is the starting quantity. In our exercise, it represents China's initial population of 1.2 billion people.
• \( k \) is the decay rate, governing how quickly numbers fall. A small \( k \) results in a slower decay.
• \( t \) denotes time, measured from a specific starting point.

Using exponential functions allows us to model realistic scenarios efficiently. As seen, by substituting the given values into the equation, we determine that despite a 0.5% annual decrease, China's population won't reach the intended 700 million within 50 years from 2000. This underscores how exponential functions provide valuable predictive power in population dynamics and related fields.