At 34 degrees north latitude, the longest day of the year has 14.273 hours of daylight, and the shortest day 9.726 hours. If the equinox falls on
march 21(day 80), model the hours of the daylight by day?
To model the hours of daylight by day, we can use the concept of a sine curve. The equation for a basic sine curve is:
y = A * sin(Bx - C) + D
Where:
- A is the amplitude, representing the difference between the maximum and minimum values of the curve.
- B determines the period of the curve, representing how often the cycle repeats.
- C is the phase shift, determining where the curve starts in relation to the x-axis.
- D is the vertical shift, determining the average value of the curve.
To determine the values for A, B, C, and D, we can analyze the given information:
A: The amplitude would be half the difference between the longest and shortest day, which is (14.273 - 9.726) / 2 = 2.7735.
B: The period can be calculated using the number of days between the longest and shortest day, which is 365. Therefore, 2π / B = 365, and B ≈ 0.01721.
C: The phase shift can be found by determining how many days have passed since the equinox, which is (80 - 80) / 365 = 0.
D: The average value of the curve can be calculated by finding the average of the longest and shortest day, which is (14.273 + 9.726) / 2 = 12.9995.
Now we can construct the equation:
y = 2.7735 * sin(0.01721x) + 12.9995
This equation models the hours of daylight by day at 34 degrees north latitude.
To model the hours of daylight by day for a specific latitude, we need to understand how the length of daylight changes throughout the year. The two key factors that influence the duration of daylight are the Earth's axial tilt and its orbit around the sun.
At 34 degrees north latitude, the variation in daylight hours is primarily driven by the change in the angle at which sunlight reaches the surface. On the summer solstice (typically around June 21st), the Northern Hemisphere is tilted towards the sun, resulting in longer daylight hours. On the winter solstice (around December 21st), the Northern Hemisphere is tilted away from the sun, leading to shorter daylight hours.
To model the hours of daylight by day, we can start by determining the number of daylight hours on the summer solstice, shortest day (winter solstice), and the equinoxes. From there, we can interpolate to estimate the duration of daylight for each day in between.
Given the information provided:
- On the longest day of the year (summer solstice), there are 14.273 hours of daylight.
- On the shortest day (winter solstice), there are 9.726 hours of daylight.
- The equinox falls on March 21st (day 80).
Let's break down the steps to model the hours of daylight by day:
1. Calculate the difference in daylight hours between the shortest and longest days:
Difference = Longest day - Shortest day
Difference = 14.273 - 9.726
2. Calculate the daily change in daylight hours leading up to the equinox:
Daily change = Difference / (Equinox day - Shortest day)
Daily change = Difference / (80 - 0)
3. Model the hours of daylight for each day leading up to the equinox:
- On the shortest day (winter solstice), there are 9.726 hours of daylight.
- For each subsequent day, add the daily change in daylight hours:
Daylight hours = Shortest day + (Daily change * day number)
By following these steps, you should be able to model the hours of daylight by day leading up to the equinox on March 21st (day 80).