Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}\). Find the carrying capacity.

Step-by-Step Solution

Answer

The carrying capacity is 150.

Step 1: Understanding the Logistic Growth Model

The logistic growth model is typically represented by the function \( f(x) = \frac{L}{1 + e^{-k(x-x_0)}} \). In our given model, \( L = 150 \).

Step 2: Identifying the Carrying Capacity

In the logistic growth model, the carrying capacity \( L \) is the maximum value that \( f(x) \) approaches as \( x \to \infty \). From the equation \( f(x) = \frac{150}{1 + 8 e^{-2x}} \), we see that the carrying capacity \( L \) is 150.

Key Concepts

Carrying Capacity

Carrying capacity is an essential concept in the logistic growth model. It refers to the maximum population size that an environment can sustain indefinitely. In nature, factors such as food availability, habitat space, and competition for resources limit population sizes.

Within the logistic growth equation, the carrying capacity is represented by the constant \( L \). This value denotes the highest point that the population curve can approach as time progresses. In mathematical terms, when the population size reaches the carrying capacity, growth slows down due to limited resources.

In our exercise example, the logistic function is given by \( f(x) = \frac{150}{1 + 8 e^{-2x}} \), where \( L \) equals 150. Therefore, 150 is the carrying capacity in this equation. It signifies the maximum number of individuals that the environment or system can support in the long run.

Exponential Functions

Exponential functions play a crucial role in mathematical modeling, especially within growth analyses. These functions are characterized by a constant rate of growth or decay and have the general form \( f(x) = a e^{bx} \).

In the logistic growth model, exponential functions serve as the foundation for describing the initial rate of population increase. At the beginning, when resources are abundant, populations grow at an exponential rate. The presence of the exponential term \( e^{-2x} \) in our logistic growth function \( f(x) = \frac{150}{1 + 8 e^{-2x}} \) indicates how population growth initially behaves exponentially. The coefficient of \( x \) in the exponent (\(-2x\) in this case) determines the speed of this growth.

As time goes on, the impact of the exponential growth diminishes, and the function starts resembling the logistic curve rather than a simple exponential increase.

Mathematical Modeling

Mathematical modeling allows us to simulate real-world behavior through mathematical expressions and equations. It involves creating models to predict outcomes and understand relationships between different variables.

The logistic growth model is a classic example of mathematical modeling, commonly used in biology, ecology, and economics to predict how populations grow over time. By capturing key factors like carrying capacity and exponential growth, this model helps in predicting the future behavior of populations.

For instance, the logistic equation \( f(x) = \frac{150}{1 + 8 e^{-2x}} \) describes how a population grows over time considering the environmental constraints. Modeling in this way helps researchers and planners to make informed decisions about resource allocation and population management.

Functions and Graphs

Functions and graphs are pivotal in visualizing mathematical models and understanding their behavior. A function defines the relationship between two quantities, often represented as equations or graphs.

In terms of the logistic growth model, functions like \( f(x) = \frac{150}{1 + 8 e^{-2x}} \) allow us to observe population dynamics visually. The graph of this function is typically S-shaped, starting with slow growth, rising sharply, and finally leveling off as it approaches the carrying capacity.

By plotting this function, we can clearly see how factors like initial growth and limits to growth (carrying capacity) interact over time. Graphs provide an intuitive understanding of how populations evolve and how closely they adhere to theoretical models.