4. Sunspots. The activity of the sun (sunspots, solar flares, and coronal mass ejection) fluctuates cyclically. Sunspot activity can be modeled by the function \( N(t)= \) \( 57.7 \sin [0.60(t-1.43)]+58.8 \) where \( t \) is the number of years since January 1,1997 and \( \mathrm{N}(\mathrm{t}) \) is the number of sunspots observed at time \( t \). According to the model: a. What is the period (length of cycle) of sunspot activity? b. What is the maximum number of sunspots observed? c. What is the minimum number of sunspots observed? d. In what coming year is sunspot activity next expected to reach a high point?

The period of the sunspot activity can be determined from the sine function's coefficient. The general formula for the period \( T \) is given by \( T = \frac{2\pi}{b} \). In our case, \( b = 0.60 \), so: \[ T = \frac{2\pi}{0.60} \approx 10.47 \text{ years} \] This means the sunspot activity goes through a complete cycle approximately every 10.47 years. The maximum number of sunspots, as indicated by the amplitude of the sine function plus the vertical shift, can be calculated as: Max \( N(t) = 57.7 + 58.8 = 116.5 \) Thus, the maximum number of sunspots observed is approximately 116.5. The minimum number of sunspots can be found similarly. Subtract the amplitude from the vertical shift: Min \( N(t) = -57.7 + 58.8 = 1.1 \) So the minimum number of sunspots observed is approximately 1.1. To find the next peak in sunspot activity, we check when the sine function equals 1, which occurs when the argument of sine is \( \frac{\pi}{2} + 2k\pi \) where \( k \) is any integer. So we solve: \[ 0.60(t - 1.43) = \frac{\pi}{2} \] \[ t - 1.43 = \frac{\pi}{1.2} \] \[ t \approx 1.43 + 2.618 \approx 4.05 \text { (in years since 1997)} \] This corresponds to approximately the year **2001** for the first high point. The next high would be after one full cycle, around **2007.5**, so the next high point would likely occur in **2008**.