Question

Biologists are monitoring the population of rabbits in a national park. Since they have natural predators in the park, the rabbit population follows a sinusoidal model. Two months after the monitoring program begins, the rabbit population numbers 300 rabbits (the lowest amount). Fourteen months after the monitoring program begins, the rabbit population reaches its highest level of 2800 rabbits. When do the rabbits next have a population of 300? Find the amplitude, phase shift, period, vertical shift, and an equation for this model. How many rabbits will there be after 6 months?

Answer 1

To find when the rabbit population next reaches 300, determine the period of the sinusoidal model and use the general sinusoidal equation to model it. The equation for the model is y = 1250 * sin((2 * pi / 12) * (x - 2)) + 1550. After 6 months, the approximate population will be 2233.97.

Step-by-step explanation:

To find when the rabbit population next reaches 300, we need to determine the period of the sinusoidal model. The period is the time it takes for the population to complete one full cycle.

Given that the rabbit population reaches its highest level of 2800 rabbits 14 months after the monitoring program begins, we can determine the period as follows:

Period = 14 months - 2 months = 12 months

Since the population is sinusoidal, we can use the general sinusoidal equation y = A * sin(B * (x - C)) + D to model it.

From the information given, we can deduce the following:

Amplitude (A): (2800 - 300) / 2 = 1250

Phase shift (C): 2 months

Vertical shift (D): (2800 + 300) / 2 = 1550

Therefore, the equation for the model is:

y = 1250 * sin((2 * pi / 12) * (x - 2)) + 1550

To find the population after 6 months, simply substitute x = 6 into the equation:

y = 1250 * sin((2 * pi / 12) * (6 - 2)) + 1550

y = 1250 * sin((pi / 3)) + 1550

y ≈ 2233.97