In 2001 Windsor Ontario will receive its maximum amount of sunlight 15.28 hrs on Jun 21 and its least amount of sunlight 9.08 hrs on Dec 21?
a. 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.
b. On what day(s) can Windsor expect 13.5 hours of sunlight?
a. To determine an equation that can model the hours of daylight function for Windsor, Ontario, we can start by recognizing that the hours of daylight function is periodic due to the earth's revolution about the sun. The most straightforward periodic function to represent this is the sine function.
Let's assume 't' represents the number of days since January 1st. In this case, we want to find the equation for the hours of daylight function, which we can denote as D(t).
The general form of the equation for a sine function is:
D(t) = A * sin(B * (t - C)) + D
In this equation, A represents the amplitude, B represents the frequency, C represents the phase shift, and D represents the vertical shift.
From the given information, we know that the maximum amount of sunlight is 15.28 hours on June 21st (which approximately corresponds to the 172nd day of the year). This indicates that the amplitude (A) is 15.28 / 2 = 7.64 (half of the maximum amount).
To determine the frequency (B), we need to find the period of the function. The period represents the number of days it takes for the daylight function to repeat itself. In this case, it is one year, so B = 2π / 365.
Since June 21st is the day with the maximum amount of sunlight, we can assume that it represents the phase shift (C = 172).
Lastly, the vertical shift (D) represents any change in the average daylight hours for the specific location. As we do not have this information, we can assume it to be zero, so D = 0.
Putting it all together, the equation that can model the hours of daylight function for Windsor, Ontario is:
D(t) = 7.64 * sin((2π / 365) * (t - 172))
b. To determine on which day(s) Windsor can expect 13.5 hours of sunlight, we can use the equation we derived in part a. We can set D(t) equal to 13.5 and solve for t.
13.5 = 7.64 * sin((2π / 365) * (t - 172))
From here, you can use algebraic techniques to solve for t. Start by dividing both sides of the equation by 7.64, then use inverse sine function (also known as arcsine or sin^(-1)) on both sides to isolate (2π / 365) * (t - 172). Finally, solve for t by rearranging the equation.
Once you have the value for t, you can convert it into a specific date by adding the number of days to January 1st. For example, if t = 100, it means 100 days have passed since January 1st, so the corresponding date would be January 1st + 100 days = April 11th in this case.
Using this approach, you can determine on which day(s) Windsor can expect 13.5 hours of sunlight throughout the year.