Problem 36 - Algebra 2

Question

For Exercises \(36-38,\) use the following information. The population of predators and prey in a closed ecological system tends to vary periodically over time. In a certain system, the population of owls \(O\) can be represented by \(O=150+30 \sin \left(\frac{\pi}{10} t\right)\) where \(t\) is the time in years since January \(1,2001 .\) In that same system, the population of mice \(M\) can be represented by \(M=600+300 \sin \left(\frac{\pi}{10} t+\frac{\pi}{20}\right)\) Find the maximum number of owls. After how many years does this occur?

Step-by-Step Solution

Answer

The maximum number of owls is 180, occurring 5 years after January 1, 2001.

Step 1: Identify the Function

The population of owls is given by the function \( O(t) = 150 + 30 \sin \left(\frac{\pi}{10} t\right) \). We need to find the maximum value of this function and determine when this occurs.

Step 2: Determine Maximum of Sine Function

The sine function \( \sin(x) \) achieves its maximum value of 1. Therefore, the maximum value of \( 30 \sin \left(\frac{\pi}{10} t\right) \) is \( 30 \times 1 = 30 \).

Step 3: Calculate Maximum Population

Add the maximum value of the sine component to the constant in the owl population function: \( O = 150 + 30 \times 1 = 180 \). Thus, the maximum number of owls is 180.

Step 4: Solve for Time When Maximum Occurs

The sine function reaches its maximum at \( \frac{\pi}{10} t = \frac{\pi}{2} + 2n\pi \), where \( n \) is an integer. Solve for \( t \) when \( n = 0 \): \( \frac{\pi}{10} t = \frac{\pi}{2} \), which simplifies to \( t = 5 \).

Step 5: Conclusion

The maximum number of owls is 180, and this occurs 5 years after January 1, 2001, which would be January 1, 2006.

Key Concepts

Sine Function

The sine function is a fundamental component of trigonometry. It is commonly represented as \( \sin(x) \). This function maps angles to values between -1 and 1. These values have varying applications in mathematics, especially in modeling oscillating systems. In the context of population dynamics, the sine function helps represent how populations increase and decrease over time.
Understanding the sine function is essential for identifying the maximum or minimum points of systems described by trigonometry. In simple terms, whenever the sine value equals 1, it is at its peak. Similarly, when it equals -1, it is at its lowest. In our exercise, we were tasked with identifying when the owl population, modeled by a sine function, reaches a maximum. When applied to sine functions like \( O(t) = 150 + 30 \sin \left(\frac{\pi}{10} t\right) \), the maximum occurs when \( \sin \left(\frac{\pi}{10} t\right) = 1 \), leading to a peak population.

Ecological System

An ecological system, or ecosystem, is a biological environment consisting of all the organisms living in a specific area and their physical environment. It's a community of living species in conjunction with non-living components. The interactions within an ecosystem can influence the population dynamics of species living therein.
In our example, the ecosystem includes predators like owls and prey such as mice. Their populations are interconnected; changes in population size for one species can affect the other. Ecosystems are often complex, with various factors influencing the state and stability of populations within them. Key factors in an ecosystem include food sources, habitat conditions, and environmental changes. They all impact how populations such as those of owls and mice, interact over time.

Population Dynamics

Population dynamics refers to the study of how and why populations change in size and structure over time. It accounts for variables such as birth rates, death rates, and migration. This field evaluates how these factors, along with environmental pressures, impact a species' growth.
The periodic rise and fall observed in predator-prey relationships, such as those of owls and mice, are classic examples of population dynamics at work. These fluctuations are often driven by the availability of food and resources, predation rates, and natural events. In a closed ecological system, external influences are limited, allowing for more predictable periodic changes, as is modeled by the sine functions in our exercise.

Periodic Functions

Periodic functions are those that repeat their values in regular intervals or periods. Examples include the sine and cosine functions, which are foundational in trigonometry. In the context of our exercise, the owl and mouse populations are modeled using periodic functions, indicating repeated cycles over time.
An important characteristic of periodic functions is their amplitude and period. Amplitude refers to the height of the peak value from the baseline, while the period is the time it takes for a full cycle to repeat. Examining the owl population equation, \( O(t) = 150 + 30 \sin \left(\frac{\pi}{10} t\right) \), reveals that the population follows a periodic trend, peaking every few years. These patterns are significant in demonstrating how populations in a stable ecological system might behave consistently over time.