Problem 24 - College Algebra

Question

Use a graphing calculator and this scenario: the population of a fish farm in \(t\) years is modeled by the equation \(P(t)=\frac{1000}{1+9 e^{-0.6 t}}.\) The formula for an increasing population is given by \(P(t)=P_{0} e^{r t}\) where \(P_{0}\) is the initial population and \(r>0 .\) Derive a general formula for the time \(t\) it takes for the population to increase by a factor of \(M .\)

Step-by-Step Solution

Answer

The time \( t \) is given by \( t = \frac{\ln(M)}{r} \).

Step 1: Understanding the Given Model

We are given a particular model for the population of a fish farm \( P(t)=\frac{1000}{1+9 e^{-0.6 t}} \). This is a logistic growth model, which differs from the exponential growth model \( P(t)=P_{0}e^{rt} \). Our task is to find a general formula for the time \( t \) when the population increases by a factor \( M \), using the exponential growth model principles.

Step 2: Expressing Population Increase by Factor M

In the exponential growth model \( P(t)=P_{0}e^{rt} \), to increase the population by a factor of \( M \), we set \( P(t) = M \cdot P_{0} \). Thus, our equation becomes \( M \cdot P_{0} = P_{0}e^{rt} \).

Step 3: Isolating the Exponential Term

Simplify the equation \( M \cdot P_{0} = P_{0}e^{rt} \) by dividing both sides by \( P_{0} \), assuming \( P_{0} eq 0 \). We get \( M = e^{rt} \).

Step 4: Solving for Time t

Take the natural logarithm of both sides of \( M = e^{rt} \) to solve for \( t \): \[ \ln(M) = rt \].Divide by \( r \) to isolate \( t \): \[ t = \frac{\ln(M)}{r} \].This formula represents the time it takes for the initial population to increase by a factor \( M \) under exponential growth conditions.

Key Concepts

Logistic Growth

Logistic growth is a concept in mathematical modeling that describes a population's growth rate which is initially exponential but eventually slows down as the population reaches its carrying capacity. This kind of growth model is more realistic for biological populations than simple exponential growth.

• In logistic growth, the population expands rapidly at first.
• The growth rate decreases as resources become limited.
• Eventually, the population size stabilizes at its carrying capacity.

The logistic growth model is represented by the function \( P(t) = \frac{K}{1+Ae^{-rt}} \) where:

• \( K \) is the carrying capacity of the environment.
• \( A \) reflects initial conditions that affect the early growth rate.
• \( r \) is the growth rate.

Understanding this model can help in planning for sustainable development, conservation efforts, and resource management in ecosystems.

Population Model

A population model is a mathematical representation of how a population changes over time. These models help scientists and researchers understand population dynamics by using equations that predict future population trends.

• The simplest model is exponential growth: \( P(t) = P_0 e^{rt} \).
• However, in reality, populations are often limited by resources and tend to follow logistic growth.
• Models can include birth rates, death rates, immigration, and emigration.

By using these models, we can explore different scenarios, such as resource depletion or changes in growth rates due to environmental factors. This helps forecast future conditions, aiding policy-making and management strategies.

Graphing Calculator

Using a graphing calculator can be invaluable in visualizing complex mathematical models such as exponential and logistic growth. These calculators allow users to input equations and immediately see the resulting graphs, which can provide a clear picture of population dynamics.

• Graphing calculators help compare different growth models by plotting them on the same graph.
• They aid in locating specific points like the carrying capacity and inflection points in logistic models.
• The technology allows for adjustments in parameters to see how changes affect the model.

Mastery of a graphing calculator enhances understanding of growth trends and supports a deeper comprehension of mathematical concepts, making them an essential tool in studying population models.