SOLUTION: The population of deer living in an enclosed game reserve displays regular fluctuations in size. Wildlife biologists have chosen to model the population size with a sinusoidal fun

Question 32568: The population of deer living in an enclosed game reserve displays regular fluctuations in size. Wildlife biologists have chosen to model the population size with a sinusoidal function. At its height the deer population is 7000, while at its low it is 2000. The time between highs and lows is 6 months. The population at the start of January, 2001 is 4500 and decreasing. Consider January, 2001 to correspond to time t = 0.
a) Model the deer population as a function of time t in months.

I'm just having a little trouble with the actual function for this one, I managed to figure out the graph I think but can't figure out how to set up the function with the numbers given. Please help if you can...
Thanks a bunch, you guys are always such a big help.
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Equation you want is y=Asin(BX+C)+D
amplitude = A = (7000-2000)/2=2500
period = 6 mths = (2pi/B); so B=pi/6
Shift = -C/B=-C/[pi/6]; shift is 6 mths to the left;
So -C/[pi/6]=+6 and C=-pi
Translation is +4500; so D=4500
Equation is y=2500sin[(pi/6)t-pi]+4500
Cheers,
Stan H.

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