Problem 389 - Precalculus

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 \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\) Graph the function.

Step-by-Step Solution

Answer

Graph the logistic function on a calculator by entering the equation and adjusting the window for clear visualization.

Step 1: Understand the Function

The function given is a logistic function: \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\). It models population growth and is characterized by an initial rapid increase that slows as the population reaches a maximum limit.

Step 2: Set Up the Graphing Calculator

Turn on the graphing calculator and access the function plotting section. Ensure you're in the correct mode (normally function or standard mode).

Step 3: Enter the Function

Input the function \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}\) into the calculator. Use the keypad to input variables, constants, and operations accordingly.

Step 4: Adjust the Viewing Window

Set the viewing window for appropriate values of \(t\) and \(P(t)\). Commonly, set \(t\) between 0 and 10, and \(P(t)\) between 0 and 1000, to see the function's behavior clearly.

Step 5: Graph the Function

Press the graph button to plot the function. Observe how the population starts close to 0, rapidly increases, and then levels off, illustrating a carrying capacity typically seen in logistic growth models.

Key Concepts

Understanding Population Growth

In biology, **population growth** refers to the increase in the number of individuals in a population. A logistic function like the one given in the exercise often describes such growth. Initially, the population grows rapidly because there are plenty of resources available and few individuals competing for them. This is called exponential growth.

However, as the population continues to grow, the rate of growth slows because resources become limited, leading to increased competition among individuals. This slower growth period is described by the logistic function as it reaches a plateau. It simulates the real-world scenario where a population can't grow indefinitely due to resource limitations. This plateau is known as the carrying capacity, where the population stabilizes. Using a logistic function is an excellent way to mathematically model this biological concept.

Utilizing a Graphing Calculator

**Graphing calculators** are powerful tools for visualizing mathematical functions. To accurately graph a logistic function like the one in this exercise, follow these steps:

• First, make sure your graphing calculator is in the correct mode. Most calculations require the function mode.
• Input the logistic function by accessing the appropriate menu or function entry option.
• Set the _viewing window_ to appropriate ranges to capture the behavior of the function. For this problem, set the time \( t \) from 0 to 10 to observe the initial growth phase clearly.
• Graph the function by pressing the graph button. This allows you to visually understand how the population grows over time.

Graphing calculators not only help in plotting functions like these but also aid in understanding concepts like the carrying capacity and where it occurs on the graph.

Exploring Carrying Capacity

**Carrying capacity** is a fundamental concept in ecology representing the maximum population size that a particular environment can sustain over the long term. For the logistic equation in the exercise, the carrying capacity is indicated by the plateau in the population graph.

The logistic function mathematically describes how growth slows down as the population nears its carrying capacity, due to factors like limited food, space, and other resources. In our fish farm scenario, this function models how the fish population will grow swiftly but eventually level off at a sustainable number, which is 1000 in this model.

Understanding carrying capacity helps in resource management and planning so that environments are not over-exploited. Observing where the graph flattens out helps in identifying this carrying capacity visually, offering insight into the dynamics of the ecosystem in question.