Question

In New York City at the spring equinox, there are 12 hours 8 minutes of daylight. The longest and the shortest days of the year vary by 2 hours 53 minutes from the equinox. In this year, the equinox falls on March 21.

A. Identify the independent and dependent variables.
B. Find the amplitude and the period of the function.
C. Create a trigonometric function that describes the hours of sunlight for each day of the year.
D. Use the function you build in part C to find out how many fewer daylight hours February 10 will have than March 21. You may look at a calendar.

Answer 1

The independent variable is the date of the year, and the dependent variable is the number of hours of daylight. The amplitude is half the difference between the longest and shortest days, and the period is one year. The trigonometric function that describes the hours of sunlight can be represented as y = A * sin((2π/365)(x - c)) + b. To find the difference in daylight hours, evaluate the function for the desired dates.

Step-by-step explanation:

The independent variable in this problem is the date of the year, while the dependent variable is the number of hours of daylight.

The amplitude of the function represents half the difference between the longest and shortest days of the year, which in this case is (2 hours 53 minutes)/2 = 1 hour 26.5 minutes. The period of the function is one year, as it takes one year for the cycle of daylight hours to repeat.

A trigonometric function that describes the hours of sunlight for each day of the year can be represented by the equation:

y = A * sin((2π/365)(x - c)) + b

Where A is the amplitude, c is the horizontal shift (representing the day of the year when the longest day occurs), and b is the vertical shift (representing the average daylight hours at the equinox).

To find out how many fewer daylight hours are on February 10 compared to March 21, you would evaluate the trigonometric function for those dates and find the difference.