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 15.520 and the number of hours of sunlight on the winter solstice is 8.484. 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 -) + B that fits the data. A = 3.518 omega = 2 pi/365

Solution

y = Asin(wt - phi) +B

A = 3.518

mean value , B = 12

y = 3.518sin(2pi*t/365 - phi) + 12

x = 172 ; y = 15.520

15.520 = 3.518sin(2pi*t/365 - phi) + 12

1 = sin(2pi*t/365 - phi)

phi = 2pi*t/365 - pi/2

t= 172

So, phi = 0.94pi - 0.5pi =0.44pi

So, y = 3.518sin(2pi*t/365 - 0.44pi) + 12