Create a equation for both.

Question

A population of rabbits oscillates 19 above and below an average of 94 during the year, hitting the lowest value in January (t = 0). Find an equation for the population, P, in terms of the months since January, t.

Answer 1

Let

A = 19 be the amplitude of oscillation

m = 94 be the average

T = 12 be the period

The general form is

P(t) = m + A sin(2πt/T + φ)

The only thing remaining is to determine the phase φ so that

minimum occurs at t = 0, while minimum for sin(x) occurs at x = -π/2.

Therefore φ = -π/2

The final result is

P(t) = 94 + 19 sin(2πt/12 - π/2)