Trig modeling - population

Question

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.

Answer 1

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:

Answer 2

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?