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A population drops from 200,000 in 1950 to 76,000 in 1996, and has risen since then. Taking into account that the population follows a sinusoidal cycle affected by environmental conditions and predation, and the population will reach its previous high again, what is a possible sinusoidal formula to describe the population as a function of time in years?

So far, I have let t=0 in 1950 A(t)= 200,000(r)^t

I'm not sure how to factor in the sinusoidal cycle part.

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2 Answers 2

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Suppose that the maximum value is $200$ (in thousands), and the minimum is $76$ (in thousands). Then we want an amplitude of $\frac{200-76}{2}$. We really want a period of $2\cdot46$ (max/min occurs at half the period), and so the coefficient of $t$ set as $\frac{2 \pi}{2 \cdot 46}$ satisfies this. A cosine function will work well here. What we have so far is $$\frac{(200 - 76)}{2}\cos\left(\frac{2\pi}{2\cdot46} t\right).$$ Now we need to add a constant, because this thing has a peak at $\frac{200-76}{2}=62$. We want it to peak at $200$, and so we add $138$, and end with $$f(t)=\frac{(200 - 76)}{2}\cos\left(\frac{2\pi}{2\cdot46} t\right)+138.$$ Here is a plot of that effect:

enter image description here

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  • $\begingroup$ thank you! That makes sense. Is there any way to tell in how many years will the population reach 200 (thousand) again? $\endgroup$
    – user112533
    May 19, 2014 at 2:52
  • $\begingroup$ Absolutely. You can solve $f(t)=200$ for $t$. The thing is, it reaches that value every $2n \cdot 46$ years from year $t=0$, and so there is really nothing to solve. At $92$ years from year $0$ the model reaches peak again. :) $\endgroup$
    – J. W. Perry
    May 19, 2014 at 2:55
  • $\begingroup$ Ah, that was very straightforward and obvious. Thank you so much for the clarity in your explanation and all of your help :) $\endgroup$
    – user112533
    May 19, 2014 at 3:01
  • $\begingroup$ My pleasure :). $\endgroup$
    – J. W. Perry
    May 19, 2014 at 3:02
  • $\begingroup$ Excuse me but I have one last question. If there were a separate population that rises when this population is under 100,000 then what percentage of the time would that separate population be higher? $\endgroup$
    – user112533
    May 19, 2014 at 5:03
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Hint:

Hint

So if the period of the $y=cos({2x\pi\over t})$ is $t$, what do you need the period of your function to be? How far horizontally do you need to shift your function? How far vertically? How much do you need to scale it?

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