A population of rabbits oscillates 32 above

Question

A population of rabbits oscillates 32 above and below average during the year, hitting the lowest value in January (t = 0). The average population starts at 650 rabbits and increases by 190 each year. Find an equation for the population, P, in terms of the months since January, t.

 = 

Answer 1

Hi Grace M.,

For the growth of the rabbits without the oscillation, with 650 initial rabbits and 190 additional rabbits annually, the monthly growth is, 650 + 190*t/12 (linear).

For the oscillation (assuming an annual oscillation) we can use the cos() function with an Amplitude = 32, a reflection (-) since it starts at it's lowest point in January, and a period of 12, to get -32cos(2π*t/12) = -32cos(π*t/6).

We can add this to the linear growth to get a final oscillation equation of:

P(t) = 650 + 190*t/12 - 32cos(π*t/6).

Go to desmos.com and graph it.

I hope this helps, Joe.