Problem 23 - College Algebra

Question

For the following exercises, use this scenario: The population \(P\) of an endangered species habitat for wolves is modeled by the function \(P(x)=\frac{558}{1+54.8 e^{-0.42 x}},\) where \(x\) is given in years. How many wolves will the habitat have after 3 years?

Step-by-Step Solution

Answer

The habitat will have approximately 34 wolves after 3 years.

Step 1: Understand the Function

The function given is \(P(x) = \frac{558}{1+54.8 e^{-0.42 x}}\). This models the population \(P\) of wolves over time \(x\) in years.

Step 2: Substitute the Value of x

We need to find the population after 3 years. To do this, substitute \(x = 3\) into the function. The expression becomes: \[ P(3) = \frac{558}{1+54.8 e^{-0.42 \times 3}} \].

Step 3: Calculate the Exponent

Calculate the exponent in the equation. Compute \(-0.42 \times 3 = -1.26\).

Step 4: Evaluate Exponential Term

Compute the exponential term \(e^{-1.26}\). This gives a value of approximately 0.2836.

Step 5: Compute Denominator

Substitute the approximate value of the exponential term into the denominator: \[ 1 + 54.8 \times 0.2836 = 1 + 15.53128 \approx 16.53128 \].

Step 6: Divide Numerator by Denominator

Now calculate the population \(P(3)\) by dividing the numerator \(558\) by the computed denominator: \[ P(3) = \frac{558}{16.53128} \approx 33.78 \].

Key Concepts

Exponential Functions

Exponential functions are a type of mathematical expression that feature an exponent. This exponent often involves variables and helps describe how quantities increase or decrease exponentially over time. Key aspects of exponential functions include:

• The base of the exponent, often denoted as "e" in scientific contexts, which approximately equals 2.718. This is known as Euler's number, a fundamental constant in mathematics.
• Exponential growth or decline, which occurs when the rate of change is proportional to the current amount, leading to rapid increases or decreases.
• The general form of an exponential function is often written as:
  \[ f(x) = a \, e^{bx} \] where "a" is the coefficient and "b" is the rate of growth or decay.

Understanding how these functions work can help explain many natural processes, such as population dynamics, where the rate of growth or decay changes constantly over time. This is key in modeling scenarios like our wolf population, where population trends are critical for conservation efforts.

Endangered Species

Endangered species are animal or plant species considered at risk of extinction in the near future, often due to human activities or environmental changes. Factors threatening these populations include habitat destruction, climate change, hunting, and pollution. Why is this important?

• Conservation: Protecting endangered species is crucial for maintaining biological diversity, resilience of ecosystems, and overall environmental health.
• Indicator Species: Many endangered species are indicators of the health of the ecosystems they inhabit, meaning their decline can signal larger environmental issues.
• Legal Protection: Various laws and regulations aim to protect these species, such as the Endangered Species Act in the U.S., by conserving their habitats and imposing penalties for harm.

In our wolf population model, the wolves are considered endangered, which emphasizes the need for accurate population modeling to inform conservation strategies and ensure their survival.

Wolf Population Dynamics

Wolf population dynamics involve studying changes in the number and distribution of wolf populations over time. These dynamics are influenced by several factors including food availability, habitat conditions, predation, and human interference. Factors Affecting Wolf Populations:

• Reproduction Rates: Wolves can reproduce quickly under ideal conditions, leading to rapid changes in population size.
• Mortality Rates: High mortality can occur due to diseases, hunting, or competition for resources, impacting growth.
• Migration and Dispersal: Wolves may move to new areas in search of food or mates, affecting local population densities.

Modeling these dynamics using mathematical functions allows scientists to predict future changes and implement effective conservation measures. In the function provided, understanding these dynamics helps monitor the wolf population, ensuring it remains sustainable and does not fall into further decline.