Problem 97 - College Algebra with Modeling and Visualization

Question

Suppose that an insect population in millions is modeled by \(f(x)=\frac{10 x+1}{x+1}\) where \(x \geq 0\) is in months. (a) Graph \(f\) in \([0,14,1]\) by \([0,14,1] .\) Find the equation of the horizontal asymptote. (b) Determine the initial insect population. (c) What happens to the population over time? (d) Interpret the horizontal asymptote.

Step-by-Step Solution

Answer

The initial insect population is 1 million, and over time it stabilizes at 10 million insects.

Step 1: Graph the function

To graph the function \(f(x)=\frac{10x+1}{x+1}\), you can plot points for values of \(x\) ranging from 0 to 14. Calculate \(f(x)\) for each integer value of \(x\) within this range and plot these points on a Cartesian plane, then draw a smooth curve through the points.

Step 2: Find the horizontal asymptote

The horizontal asymptote of the rational function \(f(x)=\frac{10x+1}{x+1}\) can be determined by dividing the coefficients of the highest degree terms in the numerator and denominator. Since the numerator and denominator are both linear (degree 1), the horizontal asymptote is \(y = \frac{10}{1} = 10\).

Step 3: Determine the initial population

The initial population is the value of \(f(x)\) when \(x=0\). Substituting into the function, \(f(0) = \frac{10(0)+1}{0+1} = \frac{1}{1} = 1\) million insects.

Step 4: Analyze the population over time

As \(x\) increases, the function \(f(x)\) approaches its horizontal asymptote, \(y = 10\). This means the population increases initially but starts stabilizing and approaches a constant value of 10 million insects over time.

Step 5: Interpret the horizontal asymptote

The horizontal asymptote \(y = 10\) implies that over a long period, the insect population will stabilize around 10 million insects. This represents the carrying capacity of the environment for the insect population.

Key Concepts

Horizontal Asymptote

In rational functions, horizontal asymptotes can provide valuable insights into the behavior of a function as the input grows indefinitely. For the function \(f(x)=\frac{10x+1}{x+1}\), the concept of horizontal asymptote comes into play when determining what value the function approaches as \(x\) becomes very large.
To identify the horizontal asymptote, we look at the highest degree terms of the polynomial in both the numerator and the denominator. Here, both are degree 1 since the highest power of \(x\) is 1. By dividing the coefficients of these terms, \(10\) in the numerator and \(1\) in the denominator, we determine that the horizontal asymptote is \(y = 10\).
As a rule of thumb:

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \(y = 0\).
• If they're equal, the horizontal asymptote is the ratio of the leading coefficients.
• If the numerator's degree is higher, there is no horizontal asymptote.

Initial Value Problem

The initial value of a function in a model can tell us about the starting condition of the system we're observing. For this insect population model with \(f(x)=\frac{10x+1}{x+1}\), finding the initial population means evaluating the function at \(x=0\). This is because \(x\) represents the time in months, and \(x=0\) refers to the beginning of the observation period.
After substitution, we get \(f(0) = \frac{10 \times 0 + 1}{0 + 1} = 1\). This initial value of 1 million insects gives us a starting point of the population model. It provides a baseline for understanding how the population evolves over time, while also emphasizing the significance of setting appropriate initial conditions in such mathematical models.

Population Modeling

Population modeling such as with this given function \(f(x)=\frac{10x+1}{x+1}\), is crucial for understanding how populations change and stabilize over time. Through rational functions, these models can indicate different rates of growth and stability, using mathematical concepts to predict real-world dynamics.
In this case, the function describes an insect population's growth as affected by time \(x\). Initially, the population grows quickly. Yet, as months pass, the rate of change slows down and approaches a horizontal asymptote. Hence, the model suggests that instead of growing indefinitely, the population will tend to stabilize, reflecting a balance between factors such as birth rates, death rates, and possibly environmental constraints.
Population models can:

• Illustrate trends over time.
• Predict long-term stability or changes.
• Help plan resource distribution or conservation efforts.

Carrying Capacity

The carrying capacity in ecological models represents the maximum population size that an environment can sustain comfortably. In this function \(f(x)=\frac{10x+1}{x+1}\), the horizontal asymptote \(y = 10\) is indicative of such a carrying capacity.
This means, according to the model, over time, the insect population will stabilize around 10 million due to the limitations imposed by available resources, habitat space, or competition among the species.
Understanding carrying capacity is vital because:

• It suggests the limits to growth, highlighting when a population may become unsustainable.
• It provides insights into the equilibrium state of the population.
• It is crucial for environmental management, conservation, and planning efforts to ensure balanced ecosystems.