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.

Question

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

Answer 1

In the equation:
N=Asin(Bt) + C
A represents half of the difference between the maximum and minimum, sometimes called amplitude.
B is a multiplitive factor to ensure that the cycle of sine (2π) fits into the physical cycle, in this problem, 365 days.
Since the amplitude of daylight hours is (18-7)/2=5.5, we obtain A=5.5.
We need a value of B such that when t=0 and t=365, Bt becomes 0 and 2π.
So 365B = 2π, or B=2π/365.