Problem 11 - 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. What is the maximum number of mice, and when does this occur?

Step-by-Step Solution

Answer

The maximum number of mice is 4250, which occurs in June.

Step 1: Understand the Sine Function

The given function is a sinusoidal function that models a periodic population of field mice. The function is given by \(y = 3000 + 1250\sin\left(\frac{\pi}{6} t\right)\). Here, \(\sin\left(\frac{\pi}{6} t\right)\) will oscillate between -1 and 1.

Step 2: Determine the Amplitude and Vertical Shift

The amplitude of the sine function is 1250. This means the population varies by ±1250 from the midline. The midline is at \(y = 3000\). Thus, the population oscillates between \(3000 - 1250\) and \(3000 + 1250\).

Step 3: Calculate the Maximum Population

The maximum value of \(\sin\left(\frac{\pi}{6} t\right)\) is 1. Substituting this into the equation gives the maximum population as \(y = 3000 + 1250 \times 1 = 4250\).

Step 4: Determine When Maximum Occurs

Max value of \(\sin\left(\frac{\pi}{6} t\right)\) is reached when its argument \(\frac{\pi}{6} t = \frac{\pi}{2}\), \(\frac{5\pi}{6}\), etc. Solving \(\frac{\pi}{6} t = \frac{\pi}{2}\) gives \(t = 3\). So the maximum population occurs 3 months after March 1, which is in June.

Key Concepts

Amplitude and Its Role in Sinusoidal Functions

The concept of amplitude is fundamental when dealing with sinusoidal functions, such as those involving sine and cosine. In the given equation, \( y = 3000 + 1250 \sin \left( \frac{\pi}{6} t \right) \), the amplitude is represented by the coefficient 1250 in front of the sine function. Amplitude essentially describes the extent of variation from a central point. It tells us how far the maximum and minimum values of the function deviate from the average or midline value.

The midline, in this case, is \( y = 3000 \). This midline represents the average population of mice. Therefore, with an amplitude of 1250, the mouse population fluctuates from 1250 mice below to 1250 mice above this central point.

Understanding amplitude is crucial for interpreting real-world sinusoidal models, as it directly relates to the degree of fluctuation or oscillation present in the scenario being modeled. In simpler terms, amplitude can be seen as measuring the 'height' of the waves produced by the function in context, translating to the highest deviation of mice numbers from the average in their population cycles.

Understanding Trigonometric Functions

Trigonometric functions such as sine and cosine are mathematical constructs that help describe phenomena varying over periodic intervals. These functions are particularly beneficial in modeling cycles and periodic events—like the annual rise and fall in population numbers.

In the equation \( y = 3000 + 1250 \sin \left( \frac{\pi}{6} t \right) \), the sine function characterizes periodic fluctuations. The argument of the sine function, \( \frac{\pi}{6} t \), indicates how frequently the cycles occur. Specifically, this function completes a cycle when its argument progresses through multiples of \( 2\pi \). This means every 12 months, the cycle of the mouse population repeats given \( \pi/6 \times 12 = 2\pi \).

In practical terms, trigonometric functions allow us to predict changes over time. They are vital for accurately modeling phenomena like varying populations, climate patterns, or tides—each having clear and identifiable peaks and troughs resembling the oscillations typical of a sine wave.

Population Modeling and Its Practical Applications

Population modeling is an essential tool for environmental science, wildlife management, and conservation biology. By using mathematical models such as the sine function, we can predict how populations of species will change over time in a specific area. This is particularly useful for studying species with seasonal variations, like those responding to prey availability, climatic conditions, or breeding cycles.

In our example equation \( y = 3000 + 1250 \sin \left( \frac{\pi}{6} t \right) \), the model forecasts the population of field mice over monthly intervals, accounting for natural fluctuations. By calculating values at specific points, such as at maximum and minimum, we understand the impact of seasonal changes on mice population numbers.

This modeling approach not only highlights temporal variations but also supports decision-making. Resource management, habitat restoration, and conservation planning often rely on such models to estimate future needs or risks, promoting sustainable coexistence between wildlife and human activities. Models like this help conservationists anticipate critical periods when populations might be at risk and allow for timely conservation actions.