A population of field mice oscillates 32 above and below an average of 85 during the year, hitting the highest value in May (t = 4). Find an equation for the population, P, in terms of the months since January, t?

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Can someone please explain this to me?

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Answer 1 · 2018-04-08T23:50:22

#P(t) = 32sin(pi/(6" mon")t-pi/6)+83#

Use the general form for a sinusoid to describe the population #P(t):

#P(t) = Asin(Bt+C)+D#

We are told that the oscillation is about a central value of #D = 85#:

#P(t) = Asin(Bt+C)+85#

We are told that the oscillation has a peak of 32 and a trough of -32; this implies that #A = 32#:

#P(t) = 32sin(Bt+C)+85#

We know that the period is #12" mon"#

#12" mon" = (2pi)/B#

#B=pi/(6" mon")#

#P(t) = 32sin(pi/(6" mon")t+C)+83#

We are told the value of the sine function is 1 when #t = 4#

#sin(pi/(6" mon")4" mon"+C) = 1#

#2/3pi+C = sin^-1(1)#

#2/3pi+C = pi/2#

#C = pi/2-2/3pi = 3pi/6-4pi/6 = -pi/6#

#P(t) = 32sin(pi/(6" mon")t-pi/6)+83#