In 2001, Windsor, Ontario received its maximum amount of sunlight,
15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on
- Due to the earth's revolution about the sun, the hours of daylight function is periodic. Determine an equation that can model the hours of daylight function for Windsor, Ontario.
- On what day(s) can Windsor expect 13.5 hours of sunlight?
Since the phenomenon is cyclical we can model with a sinusoid. The model is `y=Asin(B(t-h))+k` where A is the amplitude; B is derived from the period, t is the time in days, h is the horizontal translation and k the vertical translation. (y=k is the midline.) y represents the amount of sunlight in hours.
A: The amplitude is `A=("max"-"min")/2` so `A=(15.28-9.08)/2=3.1`
B: `B=(2pi)/p` where p is the period. The amount of sunlight should have a period of 1 year or roughly 365 days. Then `B=(2pi)/365`
h: h is the phase shift or horizontal translation. We take t=0 to be Jan. 1. The maximum of the sine function occurs one quarter of the period away from the start. This would translate to day 91; here the maximum occurs at day 171 (June 21 is day 172 but we are taking Jan 1 to be t=0, so t=171) thus there is a phase shift of 80 days to the right.
k: k is the midline or the arithmetic mean of the maximum and minimum. So `k=(15.28+9.08)/2=12.18`
Our model is `y=3.1sin((2pi)/365(t-80))+12.18`
(b) If y=13.5:
` ` `3.1sin((2pi)/365(t-80))=1.32`
But we also have to use the other possible angle for sine:
Thus Windsor will get 13.5 hours of sunlight on day 106 and day 237.