Find an equation for the population, P, in terms of the months since January, t.

Question

A population of rabbits oscillates 33 above and below average during the year, hitting the lowest value in January (t = 0). The average population starts at 700 rabbits and increases by 170 each year. Find an equation for the population, P, in terms of the months since January, t.

Answer 1

Since the population oscillates, this implies a sinusoidal term in our function. For an oscillation with a magnitude of 33 and a period of 12 months with the min at t=0, we could use:

-33*cos(2*π*t/12) = -33*cos(π*t/6)

We use negative cosine to start at the minimum of the cosine wave. 33 out front determines our magnitude. Since cos and sin oscillate between 1 and -1 normally, multiplying by 33 changes that to oscillating between 33 and -33. Finally the factor of 2*π/12 converts time from months to radians, which allows the cosine function to interpret it correctly.

To account for the base population, we need a constant term of 700. To account for the linear growth of 170 per year, we need a term of 170*t/12. The division of 12 is to convert from months to years.

Putting it all together, we get:

P(t) = 700 + 170*t/12 - 33*cos(π*t/6)