Problem 50 - Precalculus

Question

In Exercises 49 and 50 , refer to the logistic model \(f(t)=\frac{a}{1+c e^{-k t}},\) where \(a\) is the carrying capacity. As \(k\) increases, does the model reach the carrying capacity in less time or more time?

Step-by-Step Solution

Answer

As \(k\) increases, the model reaches the carrying capacity in less time.

Step 1: Understand the Logistic Model Equation

The logistic model in question is given as \(f(t)=\frac{a}{1+c e^{-k t}}\). Here, \(a\) represents the carrying capacity, \(c\) is a constant, and \(k\) is the growth rate of the function.

Step 2: Identify the Role of Constant k

In the logistic model, \(k\) determines the rate at which the function grows towards the carrying capacity \(a\). A larger \(k\) value indicates a faster rate of growth.

Step 3: Analyze the Exponential Term

The exponential term \(e^{-k t}\) in the denominator decreases more rapidly as \(k\) increases. This means the value of \(f(t)\) approaches \(a\) more quickly when \(k\) is larger.

Step 4: Relate k to Time

Since the term \(ce^{-k t}\) approaches zero quicker with a larger \(k\), the function \(f(t)\) reaches close to the carrying capacity \(a\) in a shorter period as \(k\) increases.

Key Concepts

Carrying Capacity

In the context of a logistic model, carrying capacity is a critical concept. It is represented by the letter \(a\) in the equation \(f(t)=\frac{a}{1+c e^{-k t}}\). Carrying capacity denotes the maximum population size or amount of something that an environment can sustain indefinitely. When we apply this to population growth models, it signifies the limit that the population will eventually reach, based on available resources.
When analyzing a logistic model, it's important to note that as time \(t\) goes on, and the population is nearing its carrying capacity \(a\), the growth rate starts to slow. This creates an S-shaped curve, common to logistic growth, reflecting how growth starts exponentially, then tapers off as it nears the maximum capacity.

• Carrying capacity is a boundary or a ceiling for growth.
• It can be affected by resource availability, environmental conditions, and other factors.
• Understanding this concept helps in predicting how populations will grow and stabilize over time.

Growth Rate

Growth rate in the logistic model is a vital factor that influences how quickly a population reaches its carrying capacity. Represented by the constant \(k\) in the equation \(f(t)=\frac{a}{1+c e^{-k t}}\), this term accounts for the speed of growth relative to time. A larger value of \(k\) implies a faster initial exponential growth rate.When \(k\) increases:

• The population moves more swiftly towards the carrying capacity \(a\).
• The growth is quicker, meaning the peak population will be reached sooner.

This relationship shows how a higher growth rate compresses the timeline of population stabilization, allowing models to adjust based on the specific growth dynamics being observed or desired. Therefore, knowing \(k\) is crucial when predicting behavioral trends in logistic growth scenarios.

Exponential Function

An exponential function forms a core part of the logistic model equation, specifically in the component \(e^{-k t}\). This function is pivotal because it depicts how populations, processes, or characteristics grow or decline.The expression \(e^{-k t}\) can be broken down as follows:

• \(e\) is the base of natural logarithms, approximately equal to 2.71828.
• The negative exponent \(-k\) signifies a decline factor that slows growth as time progresses.
• As \(k\) increases, the exponential decay happens faster, pushing the model towards its carrying capacity more quickly.

Exponential functions are integral to understanding complex growth models like the logistic model, as they represent the initial rapid increase followed by a slowing as the carrying capacity is approached. This model, therefore, incorporates both biological principles and mathematical expressions to simulate real-world phenomena.