Question

The Arctic fox is common throughout the Arctic tundra. Suppose the population, \(F\) of foxes in a region of northern Manitoba is modelled by the function \(F(t)=500 \sin \frac{\pi}{12} t+1000,\) where \(t\) is the time, in months. a) How many months would it take for the fox population to drop to \(650 ?\) Round your answer to the nearest month. b) One of the main food sources for the Arctic fox is the lemming. Suppose the population, \(L,\) of lemmings in the region is modelled by the function \(L(t)=5000 \sin \frac{\pi}{12}(t-12)+10000\) Graph the function \(L(t)\) using the same set of axes as for \(F(t).\) c) From the graph, determine the maximum and minimum numbers of foxes and lemmings and the months in which these occur. d) Describe the relationships between the maximum, minimum, and mean points of the two curves in terms of the lifestyles of the foxes and lemmings. List possible causes for the fluctuation in populations.

Short answer

a) It takes approximately 9 months. b) Graphing function needed. c) Foxes: Max 1500, Min 500; Lemmings: Max 15000, Min 5000. d) Fluctuations likely due to predator-prey relationships.

Step-by-step solution

- Set the Population Function Equal to 650

We need to find out when the population of foxes equals 650. Set the given function equal to 650:\[\begin{equation}500 \sin \frac{\pi}{12} t+1000 = 650\end{equation}\]

- Solve for \sin \frac{\pi}{12} t

Subtract 1000 from both sides:\[\begin{equation}500 \sin \frac{\pi}{12} t = -350\end{equation}\]Divide both sides by 500:\[\begin{equation}\sin \frac{\pi}{12} t = -0.7\end{equation}\]

- Solve for t

Take the inverse sine of both sides:\[\begin{equation}\frac{\pi}{12} t = \sin^{-1}(-0.7)\end{equation}\]Use a calculator to find the inverse sine of -0.7, which is approximately equal to -0.775. Therefore,\[\begin{equation}\frac{\pi}{12} t \approx -0.775\end{equation}\]Multiplying both sides by \[\begin{equation}\frac{12}{\pi}\end{equation}\] yields:\[\begin{equation} t \approx -2.96\end{equation}\]Because time cannot be negative, consider the periodic nature of the sine function, which is periodic every 12 months. Thus, \[\begin{equation} t \approx 12 - 2.96 = 9 \text{ (rounded to the nearest month)}\end{equation}\]

- Define the Population Function for Lemmings

The population of lemmings is modeled by the function \[\begin{equation}L(t)=5000 \sin \frac{\pi}{12}(t-12)+10000\end{equation}\]

- Graph Both Functions

Graph the functions \[\begin{equation}F(t)=500 \sin \frac{\pi}{12} t+1000\end{equation}\] and \[\begin{equation}L(t)=5000 \sin \frac{\pi}{12}(t-12)+10000\end{equation}\] on the same axes.

- Determine Maximum and Minimum Numbers from the Graph

From the graphs, you can find the maximum and minimum points. For \[\begin{equation}F(t), \text{Maximum} = 1500, \text{Minimum} = 500\end{equation}\]. For \[\begin{equation}L(t), \text{Maximum} = 15000, \text{Minimum} = 5000\end{equation}\].

- Describe Relationships and Causes

The peaks and troughs of the fox and lemming populations can indicate predator-prey relationships. As the lemming population increases, the fox population also increases, since they have more food available. When lemming numbers decline, the number of foxes also declines due to less food availability. Environmental factors, food supply, and seasonal changes can all contribute to these population fluctuations.

Key concepts

Trigonometric Functions

In the context of population models, trigonometric functions, like sine and cosine, help describe periodic behaviors. These functions repeat their values in a predictable pattern, which makes them useful for representing populations that fluctuate over time. For example, the population of Arctic foxes in the problem is modeled using a sine function, indicating that the number of foxes rises and falls in a cyclical manner. Such periodic functions typically have the form \(A \sin(Bt + C) + D\), where:

• \(A\) is the amplitude, representing the size of the population's fluctuations.
• \(B\) affects the period, determining how quickly the population cycles occur.
• \(C\) shifts the function horizontally.
• \(D\) shifts the function vertically, representing the average population.

Understanding how these elements influence the function is crucial when analyzing real-world periodic behaviors, such as animal populations.

Inverse Trigonometric Functions

Inverse trigonometric functions help us find angles or times that correspond to specific values of trigonometric functions. In the exercise, we used the inverse sine function (\(\sin^{-1}\)) to determine when the fox population would reach 650. The steps involved converting the original sine function value into an angle, from which we then derived the time.

Here’s a simplified breakdown of the steps:

• Set the population function equal to the desired value (650).
• Solve for the sine term.
• Apply the inverse sine function to both sides to find the angle.
• Convert this angle back into the time variable.

In this case, solving \(\sin^{-1}(-0.7) \approx -0.775\), allowed us to find the time by adjusting for the periodic nature of the function. This process is essential for interpreting when certain population milestones occur within cyclic models.

Predator-Prey Relationship

One core concept in ecology is the predator-prey relationship. This describes how the population dynamics of predators and prey are interconnected. In the exercise, Arctic foxes (predators) rely on lemmings (prey) for food. The population models for both species are represented using sine functions, which shows periodic growth and decline.

Key points to understand in predator-prey relationships include:

• When prey populations are high, predator populations can increase due to the abundance of food.
• As predator populations increase, they consume more prey, which can lead to a decline in the prey population.
• A decline in prey leads to a subsequent decline in predator numbers due to limited food resources.
• This cycle continues, creating a dynamic and fluctuating balance between the two populations.

Such relationships are critical for understanding ecosystem stability and the natural regulation of species populations.

Graphing Functions

Graphing functions is a powerful way to visually analyze mathematical relationships. By plotting the population models of foxes and lemmings on the same set of axes, we can easily compare their behaviors and interactions over time.

Here are useful steps for graphing such functions:

• Identify the key characteristics of each function, such as amplitude, period, phase shift, and vertical shift.
• Choose appropriate scales for the axes to accommodate the range of values.
• Plot the trigonometric function points by calculating values for specific times (\(t\)).
• Connect the points smoothly, reflecting the periodic nature of the functions.

The graph helps us find maximum and minimum population points and the times at which they occur. For example, in the exercise, the max and min values for both foxes and lemmings (\F(t) and \L(t)\ respectively) were identified. Such visual representation aids in understanding the cyclical nature of population changes and their impact on each other.