(b) Find a sine curve that models the data (as in Example 1).

Question

For reasons not yet fully understood, the number of fingerling salmon that survive the trip from their riverbed spawning grounds to the open ocean varies approximately sinusoidally from year to year. The table shows the number of salmon that hatch in a certain British Columbia creek and then make their way to the Strait of Georgia. The data is given in thousands of fingerlings, over a period of 16 years. (Let t represent years since 1985.)

Year
1985	43	1993	56
1986	36	1994	62
1987	28	1995	57
1988	22	1996	51
1989	28	1997	45
1990	33	1998	36
1991	41	1999	28
1992	48	2000	19

Answer 1

Since this data does not EXACTLY fit a sine function, you would want to do regression curve fitting. To do so, the easiest way is to use a tool like desmos. Input the data in a table in desmos and then, in a separate line enter the equation as follows:

y₁~asin(b(x₁-c))+d

From the generic sine functionwhere "A" is the amplitude, "B" is the value that determines the period, "C" is the phase shift, and "D" is the midline or vertical shift.

and desmos will return the best values of a, b, c, and d to fit the data. It also gives you a "goodness of fit" R² value of 0.9777 (as compared to R² = 1 being a perfect fit)

I got a = 18.082, b=0.520926, etc,

Let me know if you need more of an explanation.