Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, use this scenario: The population \(P\) of an endangered species habitat for wolves is modeled by the function \(P(x)=\frac{558}{1+54.8 e^{-0.462 x}},\) where \(x\) is given in years. What was the initial population of wolves transported to the habitat?

Step-by-Step Solution

Answer

The initial population was 10 wolves.

Step 1: Understand the problem

We are given the population model \( P(x) = \frac{558}{1 + 54.8 e^{-0.462 x}} \) where \( x \) represents the number of years. We need to find the population at the start, which implies when \( x = 0 \).

Step 2: Substitute the initial value into the function

Set \( x = 0 \) in the given function to find the initial population. Substitute \( x = 0 \) into the function: \( P(0) = \frac{558}{1 + 54.8 e^{-0.462 \times 0}} \).

Step 3: Simplify the exponential term

Since \( e^{0} = 1 \), simplify this in the equation: \( P(0) = \frac{558}{1 + 54.8 \times 1} = \frac{558}{1 + 54.8} \).

Step 4: Calculate the denominator

Compute the denominator: \( 1 + 54.8 = 55.8 \).

Step 5: Divide by the calculated denominator

Calculate \( P(0) = \frac{558}{55.8} \). Divide 558 by 55.8 to get the initial population.

Step 6: Perform the division

Perform the division: \( P(0) = 558 / 55.8 = 10 \). Thus, the initial population is 10 wolves.

Key Concepts

Initial Population

The idea of an initial population is crucial in understanding how species thrive in a given environment from the starting point of observation. In population modeling, the initial population represents the number of individuals present at the beginning of a study or intervention. It's often the foundation from which future population growth is projected.
To find the initial population using an exponential model, we consider the moment when time equals zero. In mathematical terms, this is represented as substituting zero into the variable for time in the equation. When dealing with exponential growth models like our given function \[P(x) = \frac{558}{1 + 54.8 e^{-0.462 x}}\]we set \(x = 0\) to determine the initial value. This substitution then simplifies the expression to capture the initial outlook of the population count.
Initial populations are crucial for setting baseline expectations and evaluating future changes. Understanding where you start helps gauge if proposed initiatives or natural conditions are impacting populations as predicted.

Exponential Function

An exponential function is a mathematical expression that is widely used to model growth or decay processes. These functions provide a way to capture how quantities increase or decrease at rates proportional to their current value.
In our wolf population example, the exponential function is embedded in the formula \[P(x) = \frac{558}{1 + 54.8 e^{-0.462 x}}\].
Here, the term \(e^{-0.462 x}\) is the exponential component, where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. The power to which \(e\) is raised (in this case, \(-0.462x\)) determines whether we are dealing with growth (positive rate) or decay (negative rate).
The nature of exponential functions allows them to model real-world situations like population changes, financial growth, and radioactive decay.

• They are characterized by a consistent rate of increase or decrease.
• The function’s output changes multiplicatively, not linearly.

Understanding exponential functions provides insight into the long-term behavior of dynamic systems, which is essential in fields like ecology, economics, and physics.

Population Modeling

Population modeling is a powerful tool used by scientists and researchers to predict changes and dynamics in groups of organisms over time. These models can vary from simple formulas to complex simulations, depending on the factors and precision required.
In the model provided, \[P(x) = \frac{558}{1 + 54.8 e^{-0.462 x}}\]represents a specific case where the number of wolves in a habitat over time is being predicted. The formula incorporates several components:

• The numerator (558) can be interpreted as the potential carrying capacity—the maximum potential population over time.
• The exponential denominator, influenced by environmental resistance factors, controls the speed and shape of growth over time.

Population models help us understand the potential impacts of various factors such as resource availability, habitat space, or competition, and allow us to estimate future population size. They are crucial for wildlife conservation, urban planning, and resource management.
By understanding these models, decision-makers can design better management strategies and predict the outcomes of conservation efforts.