Problem 10 - Algebra 2

Question

Blotocy For Exercises 10 and \(11,\) use the following information. In a certain wildlife refuge, the population of field mice can be modeled by \(y=3000+1250 \sin \frac{\pi}{6} t,\) where \(y\) represents the number of mice and \(t\) represents the number of months past March 1 of a given year. Determine the period of the function. What does this period represent?

Step-by-Step Solution

Answer

The period is 12 months, representing one full yearly cycle of population change.

Step 1: Understanding the Function

The given function is \(y = 3000 + 1250 \sin \left(\frac{\pi}{6} t\right)\). This models a sinusoidal pattern in the population of field mice over time.

Step 2: Identifying the Standard Form

The standard form of a sine function is \(a \sin(b (t - c)) + d\). For the function \(y = 3000 + 1250 \sin \left(\frac{\pi}{6} t\right)\), we identify \(b = \frac{\pi}{6}\).

Step 3: Calculating the Period

The period of a sine function \(a \sin(b t)\) is given by \(\frac{2\pi}{b}\). Substitute \(b = \frac{\pi}{6}\) into this formula to find the period: \(\frac{2\pi}{\frac{\pi}{6}} = 2 \times 6 = 12\).

Step 4: Interpreting the Period

The period of the function is 12 months. This means that the population of field mice completes one full cycle of increase and decrease every 12 months, or one year.

Key Concepts

Sinusoidal Function

A sinusoidal function is a mathematical function that describes a smooth, periodic oscillation. It's typically represented by sine or cosine functions and is characterized by a wave-like pattern. In our context, the population model for field mice uses a sine function of the form:

• Formula: \(y = 3000 + 1250 \sin \left(\frac{\pi}{6} t\right)\)
• Here, the term \(1250 \sin \left(\frac{\pi}{6} t\right)\) represents the oscillating part of the function.
• The value \(3000\) is a constant, indicating the average or baseline population of mice.

The sinusoidal function effectively captures the periodic fluctuations in the population, echoing natural seasonal cycles experienced by animals in the wild.

Period of a Function

The period of a function is the length of one complete cycle of a periodic function. For our sinusoidal function \(y = 3000 + 1250 \sin \left(\frac{\pi}{6} t\right)\), we can calculate the period by understanding the term \(b\) in the standard sine equation \(a \sin(b (t - c)) + d\).

• Here, \(b\) is \(\frac{\pi}{6}\), and the period \(T\) is calculated using \(\frac{2\pi}{b}\).
• Substituting the given value of \(b\), the calculation goes: \(\frac{2\pi}{\frac{\pi}{6}} = 2 \times 6 = 12\).

This means the function repeats its cycle every 12 months, which matches the annual rhythm of many natural processes, like migration or breeding seasons.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena in a simplified form. Here, we use a sinusoidal function to model the fluctuation in a population of field mice across a year.

• This model simplifies complexities of population dynamics, capturing essential features like growth and decline.
• It helps predict future population sizes, showing trends that are valuable in planning and conservation efforts.

Through mathematical modeling, scientists and ecologists can assess how different factors, such as season change or environmental stress, affect animal populations.

Population Dynamics

Population dynamics refers to the changes over time in population size and structure, influenced by births, deaths, immigration, emigration, and environmental factors. In the case of the field mice model:

• The sinusoidal function depicts how population size varies cyclically over 12 months.
• It reflects natural life cycles, including seasonal breeding habits, availability of resources, and predator-prey interactions.
• Such dynamics are crucial for wildlife management and ecological studies as they inform conservation strategies and sustainability practices.

Understanding population dynamics aids in preserving biodiversity and maintaining healthy ecosystems, as it predicts how populations interact with their environment and adapt to changes.