Write up a clear and complete solution with explanations to the following set of problems.
In this problem we'll use a periodic function describe the number of hours of daylight in Grand Rapids
throughout the the year. yea In particular, I will denote time as the number of days since January 1 and H will be the
number of hours of daylight. So we want to describe a function H = f(t).
Here are some (approximate) facts:
On the vernal equinox (usually March 21,180), there are 12 hours of daylight. The same happens on the
vernal equinox (usually September 21).
On the summer equinox (June 21), there are 15 hours of daylight.
• On the winter equinox (December 21), there are 9 hours of daylight.
Pro tip: I asked Wolfram Alpha what day of the year March 21 is, and it said 81, so I got 180 days since
January 1.
1. What is the period p of f(t)? Explain.
2. What is the midline m and what is the amplitude a? Explain.
3. Give a rough sketch of how the graph of f(t) should behave on the axes below.
What i
40 80 120 160 200 240 280 320 360
that sin(kt) has the proper period as shown below.
80 120 160
360
5. How can you modify this function to have the proper amplitude? (The resulting graph is shown below.)
2015
40
80 120 160 200 240 280 320 360
6. How can you modify this function to have the proper midline? (The resulting graph is shown below.)
40
80 120 160 200 240 280 320 360
7. What is one final modification you can make to obtain f(t)? (The resulting graph is shown below.)
15
10
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8. What does your function predict for the number of hours of daylight on October 9 (of 2024, which is a leap
year)? (When you compute this should your input be in degrees or radians?)