In 2001, Windsor, Ontario will receive its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs, on December 21.
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.
amplitude = (max-min)/2
center line = (max+min)/2
period is 365 days
Figure the Julian day for June 21, and start your cosine function there.
To find an equation that can model the hours of daylight function for Windsor, Ontario, we need to consider the periodic nature of the function.
The simplest periodic function is the cosine function, which repeats itself after a certain interval. We can use the cosine function to model the variation in daylight hours throughout the year.
Let's define the variable t as the number of days since January 1st, with t=0 on January 1st. We can express the number of daylight hours on any given day as:
h(t) = A * cos(B * (t - C)) + D
Where:
- A represents the amplitude of the function, which is half the difference between the maximum and minimum hours of daylight. In this case, A = (15.28 - 9.08) / 2 = 3.6.
- B represents the frequency of the function, which determines how many periods occur in a given unit of time. Since the period is one year (365 days), we need B to complete a full cycle in 365 days. Therefore, B = 2π / 365.
- C represents the phase shift, which is the shift in the function along the time axis. We know that on June 21 (t=172), Windsor receives its maximum sunlight, so C should be 172.
- D represents the mean value of the function, which is the average hours of daylight throughout the year. In this case, D = (15.28 + 9.08) / 2 = 12.18.
Putting it all together, the equation that models the hours of daylight function for Windsor, Ontario is:
h(t) = 3.6 * cos((2π / 365) * (t - 172)) + 12.18
This equation will allow you to calculate the approximate hours of daylight for any given day in Windsor, Ontario.