A population of rabbits oscillates 15 above and below average during the year, hitting the lowest value in January (t = 0). The average population starts at 1000 rabbits and increases by 6% each month

Question

A population of rabbits oscillates 15 above and below average during the year, hitting the lowest value in January (t = 0). The average population starts at 1000 rabbits and increases by 6% each month. Find an equation for the population, P, in terms of the months since January, t.

P(t) =

Answer 1

I explain how I reach this conclusion in the attached video (though I do misspeak and say 1.6 instead of 1.06 at one point), but the solution to your question is: p(t) = 1000*(1.06)^t - 15*cos(2*π*t / 12)

The 1.06^t term is from the continuous 6% population growth every month. And the 15*cos term is to model the 15 rabbit fluctuation, with the negative in front to ensure January was the lowest. The 2pi/12 term multiplying t inside cos is to make sure that 15 rabbit fluctuation happens once a year.