Problem 63 - Applied Calculus: For Business, Economics, and the Social and Life Sciences

Question

POPULATION GROWTH It is estimated that \(t\) years from now the population of a certain country will be \(P\) million people, where $$ P(t)=\frac{30}{1+2 e^{-0.05 t}} $$ a. Sketch the graph of \(P(t)\). b. What is the current population? c. What will be the population in 20 years? d. What happens to the population in the long run?

Step-by-Step Solution

Answer

Current population: 10 million. Population in 20 years: 17.24 million. Long-term population: 30 million.

Step 1: Understand the Formula

The formula for the population is given by:\[ P(t) = \frac{30}{1+2 e^{-0.05 t}} \]This represents the population in millions as a function of time in years, \(t\).

Step 2: Sketch the Graph of \(P(t)\)

To sketch the graph, consider the behavior of \(P(t)\) as \(t\) approaches 0 and as \(t\) increases indefinitely.1. As \(t \rightarrow 0\):\[ P(0) = \frac{30}{1+2e^{-0.05 \cdot 0}} = \frac{30}{1+2} = \frac{30}{3} = 10 \]2. As \(t\) increases:\(e^{-0.05t}\) approaches 0, so \(P(t)\) approaches\[ P(t) = \frac{30}{1+0} = 30 \text{ million} \]3. Intermediate points can be calculated to plot the curve between these extremes.4. Use graphing tools or software to plot the curve and observe it is a logistic growth curve.

Step 3: Calculate the Current Population

For \(t = 0\) (current time), calculate:\[ P(0) = \frac{30}{1 + 2e^{-0.05 \cdot 0}} = 10 \text{ million people} \]

Step 4: Calculate the Population in 20 Years

Substitute \(t = 20\) into the formula:\[ P(20) = \frac{30}{1 + 2e^{-0.05 \cdot 20}} \]Simplify the expression:\[ P(20) = \frac{30}{1 + 2e^{-1}} \approx \frac{30}{1 + \frac{2}{e}} \approx \frac{30}{1 + 0.74} \approx \frac{30}{1.74} \approx 17.24 \text{ million people} \]

Step 5: Analyze Long Term Population

As \(t\) approaches infinity, the exponential term \(e^{-0.05t}\) approaches 0, making the population approach:\[ P(t \rightarrow \infty) = \frac{30}{1+0} = 30 \text{ million people} \]So, in the long run, the population will stabilize at 30 million people.

Key Concepts

Logistic Growth Model

A logistic growth model describes how a population grows rapidly at first and then slows down as it approaches a maximum limit, known as the carrying capacity. This model is useful for populations that have limitations due to resources like food, space, or other environmental constraints.
In the given exercise, the population model is defined by the equation:
\[ P(t) = \frac{30}{1+2e^{-0.05t}} \]
Where:

• \(P(t)\) is the population at time \(t\) in millions.
• 30 is the carrying capacity of the population.
• \(e^{-0.05t}\) is an exponential decay function.

As \(t\) increases, the value of \(e^{-0.05t}\) decreases. Therefore, the denominator decreases and makes \(P(t)\) grow slower as it approaches 30 million. The initial growth is rapid, but it slows down near the maximum limit, forming an S-shaped curve on a graph.

Exponential Decay

Exponential decay describes how quantities decrease rapidly at first and then slow down over time. It is the opposite of exponential growth.
In this context, the term \(e^{-0.05t}\) in the population model represents exponential decay. As time \(t\) progresses, \(e^{-0.05t}\) becomes smaller and smaller.

• At \(t = 0\), \(e^0 = 1\), so the initial value is higher, leading to faster population growth.
• As \(t\) increases, \(e^{-0.05t}\) approaches 0, slowing the growth rate of the population.

This decay impacts how quickly the population stops growing and starts stabilizing. Initially, the exponential term has a large effect but diminishes significantly over time, contributing to the logistic growth model.

Long-Term Behavior

Long-term behavior in population models shows what happens to the population as time goes on indefinitely. Here, we’re interested in what happens to our population as \(t\) goes to infinity.
Using the equation:
\[ P(t) = \frac{30}{1+2e^{-0.05t}} \]

• As \(t \rightarrow \infty\), \(e^{-0.05t} \rightarrow 0\).
• This makes the denominator approach 1, so \(P(t)\) approaches 30.
• Therefore, the population stabilizes at 30 million in the long run.

Understanding this long-term behavior is crucial in predicting the sustainable population size and for planning resources accordingly. This also demonstrates how logistic models can serve as a tool for long-term population predictions.