Problem 25 - College Algebra

Question

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. Use the intersect feature to approximate the number of years it will take before the population of the habitat reaches half its carrying capacity.

Step-by-Step Solution

Answer

Approximately 7 years.

Step 1: Define Carrying Capacity

First, identify the carrying capacity of the population. The carrying capacity is the limit at which the population can sustain itself without further growth. In the given function \( P(x) = \frac{558}{1 + 54.8 e^{-0.462x}} \), the carrying capacity is the numerator, which is 558 wolves.

Step 2: Determine Half the Carrying Capacity

Next, calculate what half of the carrying capacity would be. Since the carrying capacity is 558, half of this is \( \frac{558}{2} = 279 \). This is the population value we're interested in when finding the number of years required.

Step 3: Set Population Function Equal to Half Capacity

Set the provided population function equal to 279. This gives us the equation \( \frac{558}{1 + 54.8 e^{-0.462x}} = 279 \). We need to solve this equation to find the value of \( x \).

Step 4: Solve Algebraically for Matching Equation

For simplification, consider \( \frac{558}{279} = 1 + 54.8 e^{-0.462x} \). Dividing 558 by 279 gives 2, which simplifies the equation to \( 2 = 1 + 54.8 e^{-0.462x} \).

Step 5: Isolate the Exponential Term

Subtract 1 from both sides of the equation to isolate the exponential term: \( 1 = 54.8 e^{-0.462x} \).

Step 6: Further Simplify

Divide both sides by 54.8 to solve for the exponential expression: \( e^{-0.462x} = \frac{1}{54.8} \).

Step 7: Apply Natural Logarithm

Take the natural logarithm of both sides to solve for \( x \). This gives \( -0.462x = \ln\left(\frac{1}{54.8}\right) \).

Step 8: Solve for Years

Solve for \( x \) by dividing both sides by -0.462: \( x = \frac{\ln\left(\frac{1}{54.8}\right)}{-0.462} \). Calculate this using a calculator.

Key Concepts

Carrying Capacity

In population modeling, the term "carrying capacity" refers to the maximum population size of a species that an environment can sustain indefinitely. In other words, it's the total number of individuals an ecosystem can support without destroying it. The carrying capacity is influenced by several factors including food availability, habitat space, and environmental conditions.

In our given function for wolves, the carrying capacity is represented by the number 558. This is the maximum number of wolves that the habitat can sustain. The function ensures that as time goes on, the population of wolves approaches this number, but never exceeds it.

• Carrying capacity is a crucial concept in ecology and helps planners determine sustainable practices.
• It acts as a limit to growth, meaning populations can't grow indefinitely and must stabilize at this number.

Exponential Function

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. In biological contexts, they are particularly useful in modeling population growth and decay since they can depict how populations change over time.

The exponential function in our case is represented as \(e^{-0.462x}\). Here, \(e\) is the base of the natural logarithm, approximately equal to 2.718. The exponent \(-0.462x\) dictates the rate at which the population grows or declines.

• Exponential functions can model rapid changes, such as population spikes or declines.
• These functions are integral in fields like finance, physics, and ecology.

Understanding this function helps us predict how the wolf population will behave under existing environmental conditions.

Natural Logarithm

The natural logarithm is a specific type of logarithm that uses the constant \(e\) (approximately 2.718). It is the inverse operation of taking expotential to a certain power. Its notation is typically \(\ln \), for instance, \(\ln(x)\).

In our exercise, we use the natural logarithm to solve the equation involving the exponential function, \(e^{-0.462x}\). By applying the natural logarithm, we can isolate \(x\) and determine the number of years required for the population to reach a certain level. For example, in our solution step, we end up using \(\ln\) to simplify \(\ln\left(\frac{1}{54.8}\right)\) and solve for \(x\).

• It simplifies problems involving exponential growth or decay by turning multiplicative processes into additive ones.
• Crucial for solving real-world problems where growth rates change exponentially, such as in finance and sciences.

Endangered Species

An endangered species is one that's at a high risk of extinction in the wild. These species typically have declining population numbers due to factors like habitat destruction, climate change, and overhunting. Conservation efforts are essential to prevent extinction.

The wolf population in our scenario represents an endangered species. Scientists use mathematical models to predict population trends and develop strategies to protect them. Such models help in understanding how populations can recover or continue to decline under different circumstances.

• Efforts to save endangered species include legal protection, habitat restoration, and breeding programs.
• Understanding the concept of carrying capacity and exponential growth is vital in designing these conservation strategies.

When we apply these models, we aim to ensure endangered species can reach a self-sustaining population level.