Finding an equation ( P(t)= )

Question

A population of animals oscillates sinusoidally between a low of 200 on January 1 and a high of 800 on July 1. Graph the population against time and use your graph to find a formula for the population P as a function of time t , in months since the start of the year. Assume that the period of  is one year.

Answer 1

You want to find a sine or cosine function with the right amplitude, period, y offset, and angle offset so that it fits the data.

P(t) = asin(bt-c)+d

1) You know the y range is 800 - 200 = 2a a=300 (multiplier for the function that varies from -1 to 1)

2) You know that the avg y is (800+200)/2= d d = 500 (instead of 0 for sine and cos with no offset)

3) The period is one year or 12 months (since t is in months) You want 2π(t/T) inside the sine so that the function repeats when t = T (This will be true regardless of c). So, b = 2π/12 = π/6

4) The offset takes care of the initial condition, which is the low point of sine (this occurs at -π/2), therefore, at t = 0 you want c = π/2.

You could have chosen cosine as sine and cosine are just offset by π/2 the other way.