Problem 27 - Precalculus Mathematics for Calculus

Question

World Population The relative growth rate of world population has been decreasing steadily in recent years. On the basis of this, some population models predict that world population will eventually stabilize at a level that the planet can support. One such logistic model is $$P(t)=\frac{73.2}{6.1+5.9 e^{-0.021}}$$ where \(t=0\) is the year 2000 and population is measured in billions. (a) What world population does this model predict for the year \(2200 ?\) For \(2300 ?\) (b) Sketch a graph of the function \(P\) for the years 2000 to 2500. (c) According to this model, what size does the world population seem to approach as time goes on?

Step-by-Step Solution

Answer

For 2200: P ≈ 10.7 billion; For 2300: P ≈ 12.0 billion; As t→∞, P → 12 billion.

Step 1: Identify the year for prediction

We need to predict the population for the year 2200 and 2300. For this, solve for \( t = 2200 - 2000 = 200 \) and \( t = 2300 - 2000 = 300 \).

Step 2: Calculate population for year 2200

Substitute \( t = 200 \) into the logistic model formula: \[ P(200) = \frac{73.2}{6.1 + 5.9 e^{-0.021 \, \times 200}} \]Evaluate the expression \( e^{-0.021 \, \times 200} \) and then compute the entire expression to find \( P(200) \).

Step 3: Calculate population for year 2300

Substitute \( t = 300 \) into the logistic model formula: \[ P(300) = \frac{73.2}{6.1 + 5.9 e^{-0.021 \, \times 300}} \]Evaluate the expression \( e^{-0.021 \, \times 300} \) and compute the value of \( P(300) \).

Step 4: Analyze the formula behavior for large t

Consider what happens to the expression \( e^{-0.021 \, \times t} \) as \( t \to \infty \). It tends to zero, thus simplifying the function close to the limit \[ \lim_{{t \to \infty}} P(t) = \frac{73.2}{6.1} \].

Step 5: Graph the logistic function

Plot the function \( P(t) \) over the range \( t = 0 \) (year 2000) to \( t = 500 \) (year 2500), noting how the curve approaches a horizontal asymptote as \( t \) increases.

Step 6: Interpret the stabilization limit

Evaluate the limit \( \frac{73.2}{6.1} \) to determine the stabilizing population size as time progresses, representing the population the planet will stabilize at according to the model.

Key Concepts

World Population Prediction

Predicting world population using mathematical models is a fascinating and helpful way to understand how our population might grow over time. The logistic growth model, like the one given in the exercise, provides a formula to forecast future population sizes based on certain trends. In this scenario, the model suggests how the world population might change years from now, like in 2200 or even 2300.
To predict future populations using this model, substitute the year you're interested in as a value of \( t \). For instance, to find the population in the year 2200, you'd set \( t = 200 \) since the base year is 2000. By solving the mathematical expression, this calculation gives a numerical value for the population in billions, offering a prediction that can guide policymakers and researchers.

Relative Growth Rate

The concept of relative growth rate is pivotal in understanding how quickly or slowly populations increase over time. The growth rate measures how the population is expanding relative to its current size, often expressed as a percentage. In the logistic growth model, the relative growth rate is impacted by the coefficient of \( t \), which in our case is \( -0.021 \).
As time progresses according to the model, this rate impacts the exponential part of the formula, \( e^{-0.021 \times t} \). Because it decreases over time, the exponential term shrinks, suggesting that growth slows down. This slowing trend reflects how real-world factors, such as available resources and environmental limits, influence population expansion.

Stabilizing Population

A key insight from the logistic model is the idea of a stabilizing population level. The model assumes that population growth isn't indefinite; instead, it will slow down and eventually halt at a level the planet can sustain. This is represented as the asymptote that the logistic curve approaches but never surpasses.
According to our model, as \( t \) becomes very large, the value of \( P(t) \) approaches a constant value. In mathematical terms, this is when the exponential term in the denominator approaches zero, simplifying the function to \( P(t) = \frac{73.2}{6.1} \). This value indicates the planet's carrying capacity, or the maximum stabilized population the Earth will support in the long term.

Asymptotic Behavior

Asymptotic behavior in logistic population models provides insight into long-term population dynamics. When examining the formula \( P(t) = \frac{73.2}{6.1 + 5.9 e^{-0.021 \times t}} \), it's crucial to see how the function behaves as the time \( t \) increases towards infinity.
The term \( e^{-0.021 \times t} \) showcases asymptotic behavior because it approaches zero as \( t \) gets very large. This results in the equation essentially becoming \( P(t) = \frac{73.2}{6.1} \), which is the horizontal asymptote representing the ultimate stabilizing population size. This understanding of asymptotic behavior allows us to visualize that although populations grow, they will stabilize and not increase indefinitely, aligning with realistic limitations.