What is the population of rabbits in May?

Question

A population of rabbits is at its lowest in January (t=0) and oscillates 29 above and below an average of 100 during the year. What is the population of rabbits in May?

Answer 1

Create a function that models this. Since the values are lowest at time zero, you can use a negative cosine function:

The amplitude is 29. The midline is 100. The period is 12 months ("t" is in months)

The basic negative cosine function would be:

y = -Acos[B(t - C)] + D

"A" is the amplitude, "B" defines the period (P = 2π/B so in this case, B = 2π/12 or π/6), "C" is the phase shift (no phase shift in this case so C = 0), "D" is the midline and "t" is time in months.

So y = -29cos[(π/6)t] + 100

We will need to make an assumption that "January" means Jan 1. So "May" is May 1 or 5 months into the year so t = 5:

At t = 5: y = -29cos[(π/6)5] + 100

Plug this into your calculator making sure the calculator is set to "radians". Obviously, look for an answer that is close to the max of 129.