It is readily apparent that the length of the day is getting longer now. In this problem we will model the length of the day in minutes, L(t), as a function of the date, t, using the cosine function. The model to be considered is

L(t) = a + bcos(w(t - z)),

where the constants a, b, w, and z must be determined from data about the length of the day in San Diego. You will use the information from the Navy website:
http://aa.usno.navy.mil/data/
to obtain the data you need to solve this problem.

a. First find the longest and shortest days from the Navy website by searching for the Summer and Winter Solstices. Then use the website to generate a Table of Sunrise/Sunset data for San Diego and convert the length of the longest and shortest days to minutes. To find a, you determine the midpoint between the longest and shortest days. To find b, you compute the length of time from this midpoint to the length of time of the longest day (or alternatively find half the length of time between the longest and shortest days). To find z, you need to know how many days it is until the longest day. Finally, w relates to the period of this function. Since the period is approximately 365 days, we find w by solving 365 w = 2p. (Use 366 w = 2p in a leap year.) List these computed values and write in your Lab report the function L(t). Use Excel to graph L(t) for one year.

b. Let t = 0 correspond to January 1. Use this model to estimate the length of day on New Year's day (1/1), the Vernal or Spring Equinox (check Navy table), Independence Day (7/4), and Christmas (12/25). Using the website listed above, find the actual length of the days for the dates and determine the error in using your model to compute the length of the day and create a table of these values. Which date has the maximal error and which date has the smallest error?

c. Differentiate L(t) to find the rate of change in the length of day and write this formula. Find the dates when this rate of change is increasing and decreasing the most. Also, find the rate of change for the days in Part b.