A population of rabbits oscillates 15 above and below average during the year

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 900 rabbits and increases by 190 each year. Find an equation for the population, , in terms of the months since January, t.

 =

Answer 1

There are two parts needed for the equation.

First we need to address the growth part. The population increases by 190 rabbits per year or 190/12 rabbits on average per month. So the growth part is:

P₁(t) = 900 + (190/12)t

The oscillation part is a cosine.

Acos(Bt)

To oscillate ±15 once over a year, A=-15 and B= 2π/12 = π/6. Since January is the lowest month, need A to be -15.

P₂(t) = -15cos(πt/6)

P(t) = P₁(t) + P₂(t)

= 900+190t/12-15cos(πt/6)