We are given that there are a maximum of 19.15 hours of sunlight on June 21 (day 172) and a minimum of 5.62 hours on December 21 (day 355.) We are told that the underlying function is periodic, and we are asked to find an equation for the model and to determine the day(s) when there are 13.5 hours of sunlight.
The model will be a sinusoid (a sine function, cosine function, or some combination). Thus the equation will be
`y=asin(b(x-h))+k " or " y=acos(b(x-h))+k`
where a represents the amplitude, b the transformation for the period, h the phase shift (horizontal translation or shift left/right), and k the vertical translation (shift up/down.)
(a) The amplitude is the distance from the midline to the maximum or minimum point. It can also be found by taking one half of the distance from the maximum to the minimum. So
(b) The period p=365 days as this is a good approximation for a whole year. The coefficient b is found by taking 2pi and dividing by the period p.
(c) The phase shift depends on which function we choose as a model. Since we are given a maximum, choose cosine. Then the maximum went from 0 to 172, or a shift of 172 units right.
(d) The vertical shift takes the middle of the cosine graph (y=0) to the midline, which is the average of the maximum and minimum.
Thus, a model for the function of daylight hours by day is:
(See the attachment for the graph.)
For the second question we set the model equal to 13.5 and get 2 answers in the domain of 0<t<365: t approximately 90 days and t approximately 254 days.