In a certain city the number of hours of sunlight on the sum

In a certain city, the number of hours of sunlight on the summer solstice is 15.730 and the number of hours of sunlight on the winter solstice is 8.462. Answer the following questions. Assume that summer solstice occurs on the 172^nd day of the year, and that there are 365 days until the next summer solstice, (a) Find a sinusoidal function of the form y = A sin (omega x - phi) + B that fits the data.

Solution

assume the summer solstice is the first day of the year
difference d = 15.730 - 8.462 = 7.268

it is flactuate by d/2 = 3.634

172 is near to half of year
y = 3.634sin [(2/365)t + /2] + 12.096
or
y = -3.634cos[(2/365)(t-172)] + 12.096