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.
1) 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.
2) On what day(s) can Windsor expect 13.5 hours of sunlight?
the amplitude is (15.28-9.08)/2 = 3.1
Let's say
Jun21 is day 172
Dec21 is day 355
The period is clearly 365 days, so
y = 3.1cos(2π/365 (x-172))
Now just find x when y=13.5 and figure the dates.
To determine an equation that can model the hours of daylight function for Windsor, Ontario, we can use a sinusoidal function since the hours of daylight are periodic due to the Earth's revolution around the Sun.
The standard equation for a sinusoidal function is:
y = A * sin(B(x - C)) + D
Where:
- y represents the hours of daylight
- A represents the amplitude of the function (half the difference between the maximum and minimum values)
- B represents the frequency of the function (2π divided by the period)
- C represents the phase shift (the horizontal translation of the function)
- D represents the vertical shift (the half the sum of the maximum and minimum values)
Let's calculate the values for each parameter:
1) Amplitude (A):
The difference between the maximum and minimum values is 15.28 - 9.08 = 6.20
Therefore, the amplitude is half of this difference: A = 6.20 / 2 = 3.10
2) Frequency (B):
The period of the function can be determined by the number of days between the maximum and minimum. In this case, it is 365 days (assuming a non-leap year).
To calculate B, we divide 2π (approximately 6.28) by the period:
B = 2π / 365
3) Phase Shift (C):
Since the maximum amount of sunlight occurs on June 21 (the summer solstice) and the period of the function is 365 days, the phase shift can be calculated by determining the number of days between January 1 and June 21:
C = 172 (half of 365 + 1)
4) Vertical Shift (D):
D is the halfway point between the maximum and minimum values:
D = (15.28 + 9.08) / 2 = 12.18
Now we can construct the equation:
y = 3.10 * sin((2π/365)(x - 172)) + 12.18
To determine which day(s) Windsor can expect 13.5 hours of sunlight, we can substitute y = 13.5 into the equation and solve for x:
13.5 = 3.10 * sin((2π/365)(x - 172)) + 12.18
Now, we can solve for x by isolating the sin term:
1.32 = 3.10 * sin((2π/365)(x - 172))
sin((2π/365)(x - 172)) = 1.32 / 3.10
To find the values of x, we need to use the inverse sine (arcsine) function:
(x - 172) = arcsin(1.32 / 3.10)
x = arcsin(1.32 / 3.10) + 172
Please note that the result will be given in radians, so you may need to convert it to the appropriate date format to determine the day(s) Windsor can expect 13.5 hours of sunlight.
To determine an equation that can model the hours of daylight function for Windsor, Ontario, we will use the concept of a periodic function. In this case, the hours of daylight vary throughout the year due to the earth's revolution around the sun.
1) Equation for the hours of daylight function:
We can model the hours of daylight function using a sine function since it is a periodic function that fluctuates between maximum and minimum values. The equation for the sinusoidal function is:
y = A * sin(B(x - C)) + D,
where:
- A represents the amplitude (half the difference between the maximum and minimum values)
- B determines the period (2 * pi divided by the length of the period)
- C represents the horizontal shift (in this case, it will represent the day of the year when the hours of sunlight are maximum)
- D represents the vertical shift (in this case, it will represent the average amount of sunlight)
For Windsor, Ontario:
- The maximum amount of sunlight is 15.28 hrs on June 21, which corresponds to approximately the 172nd day of the year.
- The minimum amount of sunlight is 9.08 hrs on December 21, which corresponds to approximately the 355th day of the year.
Using this information, we can plug in the values to find the equation:
A = (15.28 - 9.08) / 2 = 3.60 (amplitude)
B = 2 * pi / (355 - 172) ≈ 0.041 (period)
C = 172 (horizontal shift)
D = (15.28 + 9.08) / 2 ≈ 12.18 (vertical shift)
Therefore, the equation that models the hours of daylight function for Windsor, Ontario is:
y = 3.60 * sin(0.041(x - 172)) + 12.18.
2) Determining the day(s) with 13.5 hours of sunlight:
To find the day(s) when Windsor, Ontario can expect 13.5 hours of sunlight, we can plug in the value of 13.5 into the equation:
13.5 = 3.60 * sin(0.041(x - 172)) + 12.18.
Solving this equation will give us the day(s) when Windsor can expect 13.5 hours of sunlight. We can use numerical methods (such as graphing or using a calculator) or solve the equation symbolically using trigonometric identities and algebraic manipulations.