Question

The number of daylight hours in a day is harmonic. Suppose that in a particular location, the shortest day of the year has 7 hours of daylight and the longest day of the year has 18 hours. Then, we can model its motion with the function N=Asin(Bt) + C where t is expressed in days and A, B > 0. Find A and B, using 365 days for one year.

I need to find what A, B, and C equal. Thanks.

Answer 1

The sum of the length of the longest day and the shortest day should be 24 hours. The mean has to be 12.

Assuming instead that the numbers are 7 and 18, as your problem states, the formula would be

D (daylight hours) =
12.5 + 5.5 sin (2 pi t/365)
where t is measured in days from the vernal equinox (June 21 in the northern hemisphere)

If you want to measure t from January 1, you will have to do a transformation and include a cosine term

Answer 2

The formula I gave is correct, but the vernal equinox (from which t is measured) is March 21.