A population of animals oscillates between a low of 1098 on January 1 (t = 0) and a high of 2344 on July 1 (t = 6).

(a) Find a formula for the population, P, in terms of the time, t, in months.

To find a formula for the population, P, in terms of time, t, in months, we need to determine the characteristics of the oscillation.

Given that the population oscillates between a low of 1098 on January 1 (t = 0) and a high of 2344 on July 1 (t = 6), we can observe that the population completes one full oscillation in 6 months.

To determine the formula, we need to consider the properties of a sinusoidal function. The general formula for a sinusoidal function is given as:

P = A + Bsin(C(t - D))

Where:
P is the population at a given time t
A is the average value of the population over time
B is the amplitude, which represents the maximum deviation from the average value
C is the frequency, which determines how many oscillations occur over a given interval
D is the phase shift, which represents the horizontal displacement of the function

In this case, the average value (A) would be the average of the high and low population values:

A = (1098 + 2344) / 2 = 1721

To determine the amplitude (B), we need to find half of the difference between the high and low population values:

B = (2344 - 1098) / 2 = 623

Since the population oscillates over a period of 6 months, the frequency (C) would be:

C = 2π / T = 2π / 6 = π/3

Since the oscillation starts at t = 0, the phase shift (D) would be 0.

Now we can put all these values into the formula and obtain the final equation for the population:

P = 1721 + 623sin((π/3)(t-0))

Simplifying further, we get:

P = 1721 + 623sin((π/3)t)

Therefore, the formula for the population, P, in terms of the time, t, in months is P = 1721 + 623sin((π/3)t).