Problem 32 - Algebra 2

Question

BIOLOCY In a certain area of forested land, the population of rabbits \(R\) increases and decreases periodically throughout the year. If the population can be modeled by \(R=425+200 \sin \left[\frac{\pi}{365}(d-60)\right],\) where \(d\) represents the \(d\) th day of the year, describe what happens to the population throughout the year.

Step-by-Step Solution

Answer

The rabbit population fluctuates between 225 and 625 over the year, peaking in mid-May and reaching a minimum in early September.

Step 1: Understand the model components

The model given is a sinusoidal function: \( R = 425 + 200 \sin\left[\frac{\pi}{365}(d-60)\right] \). Here, \( R \) represents the rabbit population, \( d \) is the day of the year, 425 is the base population level, and 200 is the amplitude, which indicates how much the population varies from the base level. The term \( \sin\left[\frac{\pi}{365}(d-60)\right] \) describes the fluctuation pattern throughout the year.

Step 2: Determine the range of the population

The maximum value of \( \sin(x) \) is 1 and the minimum value is -1. Therefore, the population of rabbits \( R \) can range from \( 425 + 200 \times 1 = 625 \) to \( 425 + 200 \times (-1) = 225 \). Hence, the rabbit population fluctuates between 225 and 625 throughout the year.

Step 3: Analyze the period and phase shift

The period of the sine function is determined from \( \frac{\pi}{365} \), which represents a full year. The phase shift is 60 days, explained by \( (d-60) \). This implies that the peak and valley of the rabbit population occur 60 days after the year's start, or around early March. The period of the sine function aligns with the year, indicating a complete cycle over 365 days.

Step 4: Describe the yearly population changes

Based on the model, the population starts increasing from day 60, reaching a maximum at approximately mid-May (around day 142), then decreases to a minimum in early September (around day 225), and returns to the average population as the cycle completes by the year-end. This sinusoidal pattern repeats annually.

Key Concepts

Amplitude

In a sinusoidal function like the one modeling the rabbit population, the amplitude is a crucial concept. It represents how much the population can oscillate from its average value. In the model given \[ R = 425 + 200 \sin\left(\frac{\pi}{365}(d-60)\right), \]425 is the base population level, and 200 is the amplitude. This means that the rabbit population can vary by up to 200 more or less than the average of 425, depending on the time of the year.

• The amplitude indicates the distance from the centerline of the graph to the peak (maximum) or to the trough (minimum).
• An amplitude of 200 means that the population can reach as high as 625 when the sine function hits its peak (1), and as low as 225 at its trough (-1).

The amplitude is especially useful for understanding the extent of seasonal variation in the rabbit population, highlighting how environmental changes can significantly affect numbers.

Phase Shift

The phase shift in a sinusoidal function indicates how much the function is horizontally shifted from its usual position. In the rabbit population model \[ R = 425 + 200 \sin\left(\frac{\pi}{365}(d-60)\right), \]the phase shift is determined by the term \((d-60)\). This means that the overall pattern of the population cycle doesn't start at the beginning of the year but is delayed by 60 days.

• A phase shift of 60 days implies that significant population events, like peaks and troughs, are postponed from their conventional starting point.
• This pattern shifts key points in the population cycle, such as the maximum population around mid-May and the minimum in early September.

Understanding phase shift can help predict seasonal events and guide decisions related to managing the rabbit population and similar biological processes.

Population Modeling

Population modeling using sinusoidal functions offers insights into natural fluctuations over time. In the model \[ R = 425 + 200 \sin\left(\frac{\pi}{365}(d-60)\right), \]the variables construct a dynamic picture of the rabbit population's yearly cycle.

• The base population of 425 provides a reference point around which the population oscillates, while the sine function component captures the nature of this variability.
• The amplitude tells us about the extent of fluctuations, while the phase shift reveals timing adjustments in seasonal patterns.

By applying this model, one can foresee changes in population levels, manage resources appropriately, and understand the broader implications of similar ecological systems. It's an excellent example of how mathematical functions can replicate real-world scenarios, reinforcing the connection between mathematics and biology.

Periodicity

Periodicity is a fundamental aspect when working with sinusoidal functions, as it dictates how frequently the function's cycle repeats. In the Rabbit population function \[ R = 425 + 200 \sin\left(\frac{\pi}{365}(d-60)\right), \]the period is determined by the coefficient of \(d\) inside the sine function.

• With the coefficient \(\frac{\pi}{365}\), the population completes one full cycle every 365 days, aligning with a calendar year.
• This means the ebb and flow of the rabbit population is predictable, repeating the same yearly pattern each cycle.

Understanding periodicity is vital as it allows for the forecasting of population highs and lows throughout the year, enabling better planning and response to ecological changes.