Problem 26 - College Algebra

Question

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 \(y\) -intercept of the logistic growth model \(y=\frac{c}{1+a e^{-x}} ?\) Show the steps for calculation. What does this point tell us about the population?

Step-by-Step Solution

Answer

The y-intercept is 100; it represents the initial fish population.

Step 1: Identify the general logistic model

The given logistic growth model is expressed as \( y = \frac{c}{1 + ae^{-x}} \). When comparing it to the specific function provided, \( P(t) = \frac{1000}{1 + 9e^{-0.6t}} \), we can identify that \( c = 1000 \) and \( a = 9 \). The exponential decay rate \( r = 0.6 \) will be used for the variable \( t \) or \( x \), depending on the context.

Step 2: Find the y-intercept

To find the \( y \)-intercept of the function \( P(t) \), we substitute \( t = 0 \) into the equation. This results in \[ P(0) = \frac{1000}{1 + 9e^{-0.6 \times 0}} = \frac{1000}{1 + 9 \times 1} = \frac{1000}{10} = 100. \] Hence, the \( y \)-intercept is 100.

Step 3: Interpret the y-intercept

The \( y \)-intercept corresponds to the initial population of the fish farm when \( t = 0 \). Thus, it tells us that at the start (when no time has passed yet), the fish farm's population is 100.

Key Concepts

Exponential Decay Rate

The exponential decay rate is a crucial component in models that describe how populations change over time. It dictates how quickly or slowly the population approaches a certain threshold as time progresses.
In our logistic growth model, the exponential decay rate is represented by the term within the exponential function, specifically, the rate coefficient by which the population grows or decays. In the given equation, \[ P(t) = \frac{1000}{1+9 e^{-0.6t}}, \]the decay rate is denoted as \(-0.6\). This indicates that the population growth rate slows down over time, decreasing exponentially at a rate influenced by this coefficient.
Why is the exponential decay rate significant? Because it helps us understand the dynamics of the population growth:

• If the rate is large (positive), the decay to stability happens faster.
• If the rate is smaller (possibly negative), the population stabilizes more slowly.

Understanding the decay rate allows us to make predictions about how long it will take for the population to reach a stable, sustainable size.

Y-intercept

The y-intercept is a vital aspect of interpreting any function, especially in the context of logistic growth models. It is the point where the graph of the function crosses the y-axis, effectively representing the initial value before any changes have occurred.
In the function that models our fish farm population, \[ P(t) = \frac{1000}{1 + 9e^{-0.6t}}, \]we find the y-intercept by setting \( t = 0 \). This gives us:\[ P(0) = \frac{1000}{1 + 9 \times e^{0}} = \frac{1000}{10} = 100. \] The y-intercept is 100, meaning that the initial population when time \( t = 0 \) is 100.
What does this tell us? The y-intercept offers a snapshot of the scenario at its inception or starting point. For students examining this graphically, this point provides a clear visualization of where the population starts out, giving context to predictions and analyses of future growth.

Population Modeling

Population modeling is an insightful tool that helps scientists and researchers understand how populations will evolve over time under various conditions. The logistic growth model is particularly useful as it incorporates the concept of a carrying capacity which is the maximum population size that an environment can sustain indefinitely.
In the function for the fish farm population: \[ P(t) = \frac{1000}{1 + 9e^{-0.6t}}, \]we see several important components working together:

• The numerator, 1000, signifies the carrying capacity, marking the upper limit the population is projected to reach.
• The denominator accounts for exponential decay based on the decay rate and initial population constraints.

This model shows the outward spread of population from a small initial size or population (as seen by the y-intercept), and how it asymptotically approaches the carrying capacity.
Why is this important? By understanding these dynamics, we can predict not only the future population size but also the time frame over which growth will occur. Population modeling thus informs decisions related to resource management, conservation efforts, and strategic planning in farming or fishing industries.