Part IV. Sunspots: Sunspots are temporary phenomena on the Sun's photosphere that appear as spots darker than the surrounding areas. They are regions of reduced surface tem- perature caused by concentrations of magnetic field flux that inhibit convection. Sunspots usually appear in pairs of opposite magnetic polarity. In the 17th century, the swiss as- tronomer, Rudolph Wolf, devised the first universal method for counting sunspots. The daily sunspot number follows a periodic cycle. Goal: Find a trigonometric model for the average annual sunspot number. Sunspot Data Year Sunspot Number 1978 92.5 1979 155.4 1980 154.6 1981 140.4 1982 115.9 1983 66.6 1984 45.9 1985 17.9 1986 13.4 1987 29.4 1988. 100.2 1989 157.6 1990 142.6 1991 145.7 1992 94.3 1993 54.6 1994 29.9 1995 17.5 1996 8.6 1997 21.5 1998 64.3 1. Sketch a graph of the data, listing the year along the horizontal axis and the sunspot number along the vertical axis. What trigonometric function does this graph resemble? 2. Assume that the data is modeled by an equation of the form y = k + Asin(B(x − h)) or y=k+ A cos(B(x - h)). Using the data and your graph from part one, estimate the following variables and explain your answers for each part. (a) Calculate the amplitude as an average of 3 different values. (b) Estimate the period and determine B. (c) Estimate k as an average. (d) Estimate h. 3. Write a model for the data in the form of either y = k + Asin(B(x - h)) or y = k+ Acos(B(xh)). 4. Plot your model from number 4 with the year listed along the horizontal axis and the sunspot number along the vertical axis. How does this compare to the graph of the actual data? 5. According to your model, what is the length of the sunspot cycle in years? Explain? 6. Use your model to predict the sunspot number for 2001. How does your prediction compare with the actual value of 111.