In 2001, Windsor, Ontario received its maximum amount of sunlight, 15.28 hrs, on June 21, and its least amount of sunlight, 9.08 hrs on December 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 model the hours of daylight function for Windsor, Ontario, we need an equation that describes the periodic nature of the function. A sinusoidal function can be used for this purpose. The equation form for a sinusoidal function is:

y = A * sin(B(x - C)) + D

Where:
- A represents the amplitude of the function, which is the difference between the maximum and minimum values.
- B is the frequency factor, which determines the period of the function. In this case, the period is 365 days, which represents the Earth's revolution around the sun.
- C is the phase shift, which represents any horizontal movement of the function. Since the maximum sunlight occurs on June 21 and the minimum sunlight occurs on December 21, the phase shift will be 0 for the June date and 182.5 days for the December date.
- D represents any vertical shift of the function. In this case, it will be the average of the maximum and minimum sunlight hours, which is (15.28 + 9.08) / 2 = 12.18 hours.

Putting all these values together, the equation for the hours of daylight function for Windsor, Ontario can be written as:

y = 3.1 * sin((2π/365)(x + 182.5)) + 12.18

b. To find the day(s) when Windsor can expect 13.5 hours of sunlight, we can substitute y = 13.5 into the equation and solve for x. Let's do that:

13.5 = 3.1 * sin((2π/365)(x + 182.5)) + 12.18

First, we subtract 12.18 from both sides:

13.5 - 12.18 = 3.1 * sin((2π/365)(x + 182.5))

Then, we divide both sides by 3.1:

1.32/3.1 = sin((2π/365)(x + 182.5))

Now, we can apply the inverse sine function (sin^(-1)) to both sides to isolate the term (2π/365)(x + 182.5):

sin^(-1)(1.32/3.1) = (2π/365)(x + 182.5)

Using a calculator, we find sin^(-1)(1.32/3.1) ≈ 0.438.

Now, multiply both sides by 365/(2π):

(365/(2π)) * sin^(-1)(1.32/3.1) = x + 182.5

Calculating the left side, we get (365/(2π)) * 0.438 ≈ 70.82. Subtracting 182.5 from both sides:

70.82 - 182.5 = x

Simplifying gives:

x ≈ -111.68

So, Windsor can expect 13.5 hours of sunlight approximately 111.68 days before June 21. To find the specific date, we would subtract this value from June 21.