Problem 17 - College Algebra with Corequisite Support

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 using a calculator to observe the S-shaped curve approaching 1000 as the upper limit.

Step 1: Understand the Function

The given function is \( P(t) = \frac{1000}{1 + 9 e^{-0.6 t}} \). This is a logistic growth model, which is often used to model population growth with an upper limit.

Step 2: Set Up the Graphing Calculator

Turn on your graphing calculator. Clear any existing equations. Set the calculator to function mode, if necessary. Ensure the window settings are appropriate for viewing the function.

Step 3: Input the Function

Enter the equation \( P(t) = \frac{1000}{1 + 9 e^{-0.6 t}} \) into the graphing utility. Typically, this will require entering the numerator (1000) and then dividing it by the denominator \(1 + 9 e^{-0.6 t}\). Use the multiplication symbol for the coefficient (9) and the exponent (\(-0.6t\)).

Step 4: Adjust the Viewing Window

To see the complete behavior of the function, set the t-window (x-axis) from 0 to a reasonable number, perhaps 10 or 20, based on your calculator's display. Set the P(t)-window (y-axis) from 0 to slightly above 1000, say 1100, to adequately display the upper limit of the population.

Step 5: Graph the Function

Press the 'Graph' button to display the graph of the function. Observe the S-shaped curve characteristic of logistic functions, starting near zero, rising steeply, and approaching the upper limit.

Key Concepts

Population Modeling

Population modeling is a fascinating area of study that helps us understand how populations change over time. In our exercise, we are looking at a fish farm's population using a mathematical model. The equation given is an example of a logistic growth model, which predicts growth that starts slowly, increases rapidly, and then slows down as it reaches a certain limit. This limit is often referred to as the "carrying capacity."

• The function provided is \( P(t) = \frac{1000}{1 + 9 e^{-0.6 t}} \). Here, \( t \) represents time in years, and \( P(t) \) represents the fish population at that time.
• The logistic model is effective for populations that have constraints, like limited space or resources.
• The exponential term in the denominator \( (9 e^{-0.6 t}) \) allows the population to grow rapidly at first and then slow down as it reaches around 1000, the carrying capacity in this scenario.

This type of modeling is crucial in ecology, environmental science, and resource management.

Graphing Calculator

Using a graphing calculator is a powerful way to visualize mathematical functions and better understand their behavior. It can help you see patterns in data and predict future outcomes, which is particularly important in population modeling.

• First, ensure your calculator is in the correct mode, like function mode, to handle equations naturally.
• Clearing previous equations is important to avoid confusion and ensure the new function is the only one graphed.
• Proper window settings are key: you need to see enough of the graph to understand the general shape and behavior.

In our exercise, adjusting the viewing window allows us to see the characteristic S-shape of logistic growth. By doing so, we can observe how the population grows over time and approaches its upper limit.

Exponential Functions

Exponential functions are a core mathematical concept often used in modeling real-world situations, such as interest growth, radioactive decay, and population growth. Understanding them is crucial for this exercise.

• In the equation \(P(t) = \frac{1000}{1 + 9 e^{-0.6 t}}\), the exponential part is \(e^{-0.6 t}\).
• The base of the natural logarithm, \(e\), is approximately 2.718 and is pivotal in calculations involving growth and decay.
• The exponent, \(-0.6 t\), means that as \(t\) increases, the term \(e^{-0.6 t}\) decreases, which impacts the rate of growth.

Exponential functions model situations where change is proportional to the current amount, making them ideal for modeling natural processes and populations. This function is part of the underlying mechanism of the logistic model, driving the initial rapid growth phase before the limit is approached.