Problem 447 - Precalculus

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 of wolves was 10.

Step 1: Understand the Problem

We are given a logistic function that models the population of wolves in a habitat over time. The function is \( P(x)=\frac{558}{1+54.8 e^{-0.462 x}} \), where \( x \) is the number of years after the wolves were introduced into the habitat. We need to find the initial population, which corresponds to \( x=0 \).

Step 2: Substitute \( x=0 \) into the Function

To find the initial population, substitute \( x=0 \) into the function. Thus, the expression for the population will become \( P(0)=\frac{558}{1+54.8 e^{-0.462 \times 0}} \).

Step 3: Simplify the Exponential

Calculate the exponential expression when \( x=0 \): \( e^{-0.462 \times 0} = e^0 = 1 \). This simplifies the population function to \( P(0)=\frac{558}{1+54.8 \times 1} \).

Step 4: Calculate the Initial Population

Substitute the simplified exponential value back into the population function: \( P(0)=\frac{558}{1+54.8} \). Then perform the arithmetic: \( P(0)=\frac{558}{55.8} \).

Step 5: Perform the Division

Divide the number 558 by 55.8 to find the initial population: \( \frac{558}{55.8} \approx 10 \).

Step 6: Verify Calculations

Double-check the division to ensure accuracy and confirm that the initial population is indeed 10 wolves.

Key Concepts

Population Modeling

Population modeling is a way to describe how populations grow, shrink, or remain stable over time. In biology and ecology, it helps scientists understand species' dynamics within an ecosystem. One method of population modeling is using mathematical functions, like the logistic function, to predict population changes.

• In our scenario, the logistic function helps us describe the wolf population in their new habitat.
• The function is essential for making predictions about how the population will change over time.
• It also allows conservationists to plan better interventions if needed.

Logistic functions are particularly useful when dealing with growth that starts rapidly and then slows down as it approaches a maximum limit, called the carrying capacity. This reflects real-world constraints like limited resources that slow down population growth as numbers increase. It provides a realistic way to predict and understand population behaviors in controlled environments.

Exponential Function

An exponential function is a type of mathematical function that describes processes that grow or shrink at a steady rate over time. It is characterized by having a constant base raised to a variable exponent. In our equation, the exponential part is represented by the term involving the mathematical constant 'e'.

The formula in the logistic function is:

• The exponential term is given as \( e^{-0.462x} \).
• This part of the function models the rate at which the population change slows down over time.
• When the exponent becomes zero, the exponential value is \( e^0 = 1 \), which simplifies calculations significantly.

This pattern shows how initial rapid growth is tempered as external factors, like resource availability, become limiting. Understanding exponential functions helps in grasping how populations grow suddenly and then decelerate as they saturate the environment's capacity.

Initial Value Problem

An initial value problem in mathematics, especially in differential equations, determines an unknown function given conditions at the starting point. In population modeling, this concept helps find how many individuals started a population or specific characteristics at the start of observation.

• In our wolf population model, we identify the initial population by substituting \( x = 0 \).
• This calculation sets the stage for understanding the model's predictive behavior over time.
• By solving the initial value problem, we provide foundational information necessary for projecting future population dynamics.

Finding the initial population helps understand both current status and future projections. Recognizing that all predictive models need reliable starting data is crucial for accurate analysis and planning in ecological management and conservation initiatives.