can someone please explain how to find this equation?!?

Question

a population of animals oscillates sinusoidally throughout the year. the population has a high of 4200 on march 1 and reaches a low of 1700 on september 1. the population returns to a high of 4200 the following march 1. find a forumal that gives the population,P, as a function of time, t, measured in months since the beginning of the year. (this means t=0 represents Jan1, t=1 represents Feb1..etc.)

Answer 1

The standard sinusoid is
y = sin(x)

We have to make few changes to fit this sinusoid to our case.

The period of standard sinusoid is 2π. The period of our sinusoid is 12. The ratio of both periods is 2π/12=π/6. So the first change is:

y = sin(πx/6)

The standard sinusoid starts at x=0. (There is no phase shift.) The first maximum is at π/2, or one quarter of the full period 2π.

But our sinusoid has a maximum not at 1st April (one quarter of full year period), but one month earlier. So the phase shift of our sinusoid is -1/12. (-1 months from 12 months period.) So the second change is:

y = sin(πx/6 + 1/12)

The standard sinusoid oscillates around the average y=0. Our sinusoid oscillates around the average y=(4200+1700)/2=29500. So the third change is:

y = 2950 + sin(πx/6 + 1/12)

The amplitude of standard sinusoid is 1. The amplitude of our sinusoid is (4200-1700)/2=1250. So the fourth (and last) change is:

y = 2950 + 1250 sin(πx/6 + 1/12)

So the answer is:

P = 2950 + 1250 sin(πt/6 + 1/12)


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