Final answer:
The sought sinusoidal function representing the field mice population as a function of time, P(t), in months since January 1 is P(t) = 5000 + 2000·sin(π(t - 2)/6), reflecting the periodic rise and fall between the 3000 and 7000 individuals over a 12-month period.
Step-by-step explanation:
The student is seeking to model the fluctuating population of field mice over time with a sinusoidal function. Here's an approach to finding such a function:
- The population reaches a minimum of 3000 on March 1 (which is 2 months after January 1) and again after a period of 12 months, i.e., the next March 1. This suggests a periodic function with a period of 12 months.
- The population reaches its maximum of 7000 on September 1, which is 6 months after the minimum value observed in March. This implies that the sinusoidal function's maximum occurs at the midpoint of the period.
- The amplitude of the fluctuation is ½(7000 - 3000) = 2000, since the population varies by this amount above and below the average value.
- The average population level is ½(7000 + 3000) = 5000, which serves as the vertical shift or the midline of the sinusoidal function.
- Since the population is at a minimum in March and sinusoidal functions usually start at a midline, we need to shift the function to the right by 2 months to make it fit our specific scenario.
A potential function P(t), representing the population of field mice t months after January 1, could therefore be expressed as:
P(t) = 5000 + 2000·sin(π(t - 2)/6)
This function depicts a sinusoidal pattern where the population of field mice fluctuates between 3000 and 7000 with a period matching the 12-month cycle outlined.