Problem 74 - A Graphical Approach to College Algebra

Question

Solve each problem. Suppose that the population of a species of fish in thousands is modeled by $$f(x)=\frac{x+10}{0.5 x^{2}+1}$$ where \(x \geq 0\) is in years. (a) Graph \(f\) in the window \([0,12]\) by \([0,12] .\) What is the equation of the horizontal asymptote? (b) Determine the initial population. (c) What happens to the population after many years? (d) Interpret the horizontal asymptote.

Step-by-Step Solution

Answer

As years increase, the population declines towards zero.

Step 1: Analyze the Function

The given function is \(f(x) = \frac{x+10}{0.5x^2+1}\). Our task is to understand its behavior by graphing it and determining specific values and asymptotic behavior.

Step 2: Graph the Function

Plot the graph of \(f(x)\) over the interval \([0, 12]\) for both x and y axes. You can use graphing software or a graphing calculator. Look for any patterns or trends at the boundaries of the interval.

Step 3: Find the Horizontal Asymptote

Compare the degrees of the numerator and the denominator. Since the degree of the denominator (2) is greater than that of the numerator (1), the horizontal asymptote is \(y = 0\).

Step 4: Determine the Initial Population

Substitute \(x = 0\) into the function to find the initial population: \(f(0) = \frac{0+10}{0.5(0)^2+1} = 10\). This means the initial population is 10,000, as the population is measured in thousands.

Step 5: Describe Long-Term Population Behavior

Analyze the function behavior as \(x\) approaches infinity. Because the horizontal asymptote is \(y = 0\), the population approaches 0 as the years pass. This indicates that the species of fish will eventually become extinct if the trend continues.

Step 6: Interpret the Horizontal Asymptote

The horizontal asymptote \(y = 0\) suggests that as time progresses, the population will decline towards zero. This indicates a decrease in the fish population over many years, potentially leading to extinction.

Key Concepts

Horizontal Asymptote

In population modeling, the horizontal asymptote helps us understand the long-term trend in a population function. For the given function \( f(x) = \frac{x+10}{0.5x^2+1} \), we need to determine the horizontal asymptote to predict how the population behaves over a long period. The horizontal asymptote is determined by comparing the degrees of the polynomial in the numerator and the polynomial in the denominator.

• If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \( y = 0 \).
• If both degrees are equal, the asymptote is the ratio of the leading coefficients.
• If the degree of the numerator is greater, there is no horizontal asymptote.

For this function, the degree of the denominator (2) is greater than the degree of the numerator (1), which gives us the horizontal asymptote \( y = 0 \). This prediction tells us that regardless of how the population starts, it will trend towards zero over time.
Understanding horizontal asymptotes in population models clarifies how populations will behave indefinitely, giving insights into potential challenges and opportunities for interventions.

Initial Population

The initial population provides us with a starting point for understanding how a population model begins its trajectory. To find the initial population of the fish in the model \( f(x) = \frac{x+10}{0.5x^2+1} \), we evaluate the function at \( x = 0 \): \[ f(0) = \frac{0 + 10}{0.5 \times 0^2 + 1} = 10 \] This means the initial population is 10,000 fish, as the population is measured in thousands. The initial state of a population is crucial for setting expectations and planning potential conservation efforts.

• It shows the starting health of a population.
• Provides a baseline for future comparisons.
• Is necessary for calculating growth rates.

An accurately measured initial population enables scientists and policymakers to monitor changes and react to unexpected shifts. In this example, understanding that there were 10,000 fish initially allows for tracking growth or decline effectively over time.

Long-Term Behavior

Long-term behavior in population models predicts what will happen to a population over an extended period. In the given model \( f(x) = \frac{x+10}{0.5x^2+1} \), we observe that as \( x \) approaches infinity, the function approaches the horizontal asymptote at \( y = 0 \). This suggests that the population of fish will eventually decline to zero, indicating extinction if current trends continue.

• Long-term predictions are critical for taking preemptive actions to conserve or manage populations.
• A declining trend may spur conservation efforts or policy changes.
• Predicting extinction risks allows for strategic interventions to support population recovery.

Understanding long-term behavior helps stakeholders make informed decisions about maintaining biodiversity and ecosystem stability. Accurate long-term modeling lets conservationists identify high-risk species and ecosystems that require immediate attention or resources.