Problem 8 - College Algebra

Question

Use the logistic growth model \(f(x)=\frac{150}{1+8 e^{-2 x}}.\) Find and interpret \(f(4)\) . Round to the nearest tenth.

Step-by-Step Solution

Answer

The value of \(f(4)\) is approximately 149.6, indicating the population size.

Step 1: Understand the Logistic Growth Model

The logistic growth model is given as a function \[f(x) = \frac{L}{1 + C e^{-kx}}\]where \(L\) is the carrying capacity, \(C\) is a constant related to the initial population size, and \(k\) is the growth rate. In this exercise, the function is \(f(x) = \frac{150}{1+8 e^{-2 x}}\) with \(L = 150\), \(C = 8\), and \(k = 2\).

Step 2: Substitute x=4 into the Function

Substitute \(x = 4\) into the logistic function:\[f(4) = \frac{150}{1 + 8 e^{-2 \times 4}}\]

Step 3: Calculate the Exponential Part

Calculate the exponential term involved:\[e^{-2 \times 4} = e^{-8}\]Using a calculator, \(e^{-8} \approx 0.000335\).

Step 4: Simplify the Denominator

Substitute \(e^{-8}\) into the denominator:\[1 + 8 e^{-8} = 1 + 8 \times 0.000335 = 1 + 0.00268 = 1.00268\]

Step 5: Calculate the Value of the Function

Substitute the simplified denominator back into the function:\[f(4) = \frac{150}{1.00268}\]Calculate using a calculator: \(f(4) \approx 149.6\).

Step 6: Round and Interpret the Result

Round \(149.6\) to the nearest tenth, which remains \(149.6\). The interpretation is that at \(x = 4\), the modeled population is approximately 149.6 individuals.

Key Concepts

Carrying Capacity

The concept of carrying capacity is fundamental to understanding logistic growth models. Carrying capacity, denoted as \( L \) in the logistic equation, represents the maximum population size that an environment can sustain indefinitely. This ceiling exists due to limitations such as resources, space, and other environmental factors. The logistic growth model reflects the balancing act between population growth and environmental limits:

• Initially, populations grow rapidly when numbers are low, but growth slows as they approach the carrying capacity.
• Unlike exponential growth, which assumes unlimited resources, logistic growth considers these constraints, leading to an \( S \)-shaped curve.
• In our specific function, \( L = 150 \), indicating that the population will not exceed 150 units regardless of intrinsic growth rates.

Understanding the carrying capacity helps predict how populations behave under confined scenarios, making it a vital component in population modeling.

Exponential Growth

Exponential growth is the process by which a quantity increases over time at a rate proportional to its current value. It is characterized by the formula \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is time.

However, in the logistic model, this pure exponential growth is tempered by the carrying capacity. Nevertheless, the exponential term \( e^{-kx} \) still plays a crucial role:

• In the formula, \( e^{-kx} \) rapidly decreases as \( x \) increases, dampening the unrestricted exponential growth.
• This term accounts for the slowing down effect observed as the population approaches the carrying capacity.
• In our example: We see \( e^{-2x} \) reduce quickly, curbing the growth as part of the logistic curve.

Thus, exponential growth helps link initial rapid increases to eventual stabilization, illustrating the transition within the logistic model.

Population Modeling

Population modeling involves mathematical representations to predict how populations change over time. The logistic growth model offers a more realistic portrayal of population dynamics as compared to unrestricted models.

A few key points about using logistic growth for population modeling include:

• It introduces the concept of equilibrium, where growth stabilizes at carrying capacity.
• This model is useful in ecology and resource management, where understanding limits is crucial.
• The insights obtained can help anticipate changes in population sizes related to policy, conservation, and environmental changes.

To interpret \( f(4) \) in our exercise, we see that the model estimates a population of approximately 149.6 when \( x = 4 \), indicating near saturation relative to the carrying capacity. Hence, logistic growth models are essential tools for translating mathematical findings into practical conservation strategies.