The summer and winter solstices are the longest day and night of the year, respectively. (please help!)

Question

The summer solstice happens between June 20 and June 21, while the winter solstice is between December 21 and December 23. At a latitude of 44° the summer solstice is 16.5 hours long and the winter solstice is 16 hours long. Write a sinusoidal equation modeling the hours of daylight in a year. Use t for the variable and measure t in weeks. For the purposes of this problem, idealize a year to be 52 weeks long, with the winter solstice a week before the new year.

Answer 1

Function Y=Asin(wt-Φ)+β

Amplitude A=(16.5-8)/2=4.25 hours

Vertical shift β= (16.5+8)/2=12.25

Period is 52 weeks, thus 52=2π/ω→ω=2π/52=π/26

Now we need to find the phase shift Φ: If we divide the whole year in 4 sub-intervals each interval would be :

[0,13], [13,26],[26,39],[39,52]

The summer solstice occur at the end of week 24.

The winter solstice occur at the end of week 51.

Maximum of sine function happens at the end of first sub-interval [0,13]

So, we need to shift the graph from 24 to 13 to the right. 24-13=11

So, Φ is 11, thus, the function would be

Y=4.25 sin[(π/26)(t-11)]+12.25

We can check to see if it is correct or not by checking the winter solstice at 51 week of the year.

Y=4.25 sin[(π/26)(51-11)]+12.25 = 8 hrs, which is the day time (hrs) in winter solstice.