Problem 22 - College Algebra

Question

For the following exercises, use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}} .\) What is the carrying capacity for the fish population? Justify your answer using the graph of \(P\).

Step-by-Step Solution

Answer

The carrying capacity is 1000 fish. The graph shows \(P(t)\) approaching 1000 as \(t\) increases.

Step 1: Understanding the Function

We have the given function for population: \(P(t) = \frac{1000}{1 + 9 e^{-0.6 t}}\). This is a logistic growth function where \(1000\) is an important parameter. Recognize that the logistic function has a carrying capacity.

Step 2: Identifying the Carrying Capacity

In a logistic growth function of the form \(P(t) = \frac{L}{1 + ae^{-kt}}\), \(L\) is the carrying capacity. In our function, it is \(1000\). Hence, the carrying capacity for the fish population is \(1000\).

Step 3: Using a Graph to Justify

Use a graphing calculator to plot the function \(P(t)\). The graph will show an S-shaped curve approaching a horizontal asymptote. Observe that as \(t\) increases, \(P(t)\) approaches \(1000\), confirming it as the carrying capacity.

Key Concepts

Carrying Capacity

In the study of population dynamics, carrying capacity represents the maximum population size an environment can sustain indefinitely. It's the upper limit of population growth that can be supported by the available resources, such as food, habitat, water, and other necessities. In the formula of the logistic function, the carrying capacity is denoted by the parameter "L."
For the logistic function used in the fish farm scenario, the carrying capacity was illustrated as 1000. This means that, regardless of other conditions, the population of the fish will not exceed this number. In real-world applications, understanding carrying capacity can help in ensuring the sustainability of an ecosystem.

• Helps in resources management
• Critical for sustainable practices
• Provides a theoretical maximum limit for populations

The concept of carrying capacity is essential in conservation biology, environmental management, and agriculture.

Graphing Calculator

A graphing calculator is a powerful tool for visualizing mathematical functions and solving equations, making it especially useful in examining logistic growth models. By entering the logistic function into a graphing calculator, one can see the changes in the population model over time. Here, we used a graphing calculator to confirm the carrying capacity of the fish farm population.
When you plot the function, the graph reveals an S-shaped curve characteristic of logistic growth. As time passes, the curve levels off, reflecting the population approaching the carrying capacity of 1000. This visual representation is crucial for:

• Understanding the behavior of complex functions
• Confirming analytical solutions by visualizing them
• Exploring changes in parameters and their effects
• Providing immediate feedback and insight into mathematical concepts

Through these features, graphing calculators become an educational aid, bridging the gap between theory and practical understanding.

Logistic Function

The logistic function is often used to model population growth where resources are limited. It is characterized by an S-shaped curve due to its initial exponential growth, followed by a slowdown as the population approaches the carrying capacity.
The general form of a logistic function is:\[ P(t) = \frac{L}{1 + ae^{-kt}} \]where:

• \(L\) is the carrying capacity
• \(a\) and \(k\) are constants that affect the shape and steepness of the curve

In our fish farm example, the function is:\[ P(t) = \frac{1000}{1 + 9e^{-0.6t}} \]highlighting that the population cannot exceed 1000, indicating the logistic growth model's functionality.
Understanding logistic functions is pivotal for anyone analyzing systems where growth is self-limiting, such as ecology, medicine (e.g., spread of diseases), and economics.

Population Modeling

Population modeling involves using mathematical frameworks to understand how populations change over time within an environment. Logistic growth, in particular, is a type of population model that is well-suited for populations where resources are limited and growth eventually plateaus.
In the case of our fish farm, population modeling via the logistic function helps us predict and manage the fish population effectively. The initial parameters can significantly influence the outcome and help in decision-making processes. By utilizing population models, we can:

• Anticipate future changes in population size
• Manage ecosystems sustainably
• Plan for conservation efforts
• Evaluate impacts of policy changes or environmental shifts

Population modeling is a versatile tool in both academic research and practical applications, aiding in the forecast and management of biological populations.