Problem 83 - Algebra and Trigonometry

Question

Population Growth Suppose that the rabbit population on Mr. Jenkins' farm follows the formula $$ p(t)=\frac{3000 t}{t+1} $$ where \(t \geq 0\) is the time (in months) since the beginning of the year. (a) Draw a graph of the rabbit population. (b) What eventually happens to the rabbit population?

Step-by-Step Solution

Answer

The rabbit population approaches 3000 as time increases.

Step 1: Graph the Function

To draw the graph of the function \( p(t) = \frac{3000t}{t+1} \), we first need to identify key points and analyze the behavior of the function. Start by finding the y-intercept, which occurs at \( t = 0 \): \( p(0) = \frac{3000 \cdot 0}{0+1} = 0 \). Note that as \( t \) increases, the function rises but never reaches a peak due to the asymptotic behavior. Plot several points for \( t = 1, 2, 3 \), etc.: \( p(1) = 1500, p(2) = 2000, p(3) = 2250 \), and so on. Then, sketch the graph, which approaches a horizontal asymptote at 3000.

Step 2: Analyze the Asymptotic Behavior

To understand what eventually happens to the rabbit population, examine the behavior of the function \( p(t) = \frac{3000t}{t+1} \) as \( t \) approaches infinity. Simplify the expression for large values of \( t \): \( p(t) \approx \frac{3000t}{t} = 3000 \). This shows that the rabbit population approaches the value of 3000.

Key Concepts

Asymptotic Behavior

The concept of asymptotic behavior is crucial when examining how functions behave as input values become very large or very small. In the case of our rabbit population, represented by the function \( p(t) = \frac{3000t}{t+1} \), we are interested in what happens as time, \( t \), approaches infinity.

As \( t \) becomes very large, the term \( t + 1 \) in the denominator becomes nearly indistinguishable from \( t \) itself. This makes the fraction \( \frac{3000t}{t+1} \) simplify to \( 3000 \). This indicates that the rabbit population levels off at 3000.

This is typical in many scientific and practical situations where growth is limited by a resource or capacity, showing the population's dependency on the asymptotic behavior of the function.

Graphing Functions

Graphing functions is an essential skill to visually understand how a function behaves over its domain. To graph the function \( p(t) = \frac{3000t}{t+1} \), begin by calculating specific values for \( t \). Plot points like:

• \( t = 0 \), \( p(0) = 0 \)
• \( t = 1 \), \( p(1) = 1500 \)
• \( t = 2 \), \( p(2) = 2000 \)
• \( t = 3 \), \( p(3) = 2250 \)

Using these points, connect them with a smooth curve. The curve should approach a line as \( t \) increases, showing how the rabbit population stabilizes over time.

The graph provides a useful tool for visualizing function behavior and predicting future population values.

Rational Functions

Rational functions are ratios of polynomial expressions. The function for the rabbit population is \( \frac{3000t}{t+1} \), which consists of a polynomial in both the numerator and the denominator.

Understanding rational functions involves examining their long-term behavior, asymptotes, intercepts, and any points of discontinuity. They are commonly used to describe real-world phenomena like growth rates and population dynamics because their ratios can easily represent how one variable affects another.

Because of the division, these functions can exhibit diverse behaviors, offering insights into stability and limitation factors present in dynamic systems like populations.

Horizontal Asymptote

A horizontal asymptote reflects the value a function approaches as \( t \) goes to infinity. For the function \( p(t) = \frac{3000t}{t+1} \), there is a horizontal asymptote at \( p(t) = 3000 \).

To find a horizontal asymptote, focus on the leading terms of the numerator and denominator as everything else becomes negligible for large \( t \). Here, as \( t \) increases, \( \frac{3000t}{t} \) simplifies directly to 3000.

This helps us understand that no matter how big the time \( t \) becomes, the rabbit population does not exceed or drop below this threshold, indicating a limiting capacity on the farm's environment. Horizontal asymptotes provide valuable insights into the long-term sustainability of such systems.