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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.

  • Math -

    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

  • Math (correction) -

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

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