The document provides information about sunrise times in Montreal throughout the year and asks questions to model this data with a sinusoidal function. It gives the latest sunrise of Dec. 22 at 9:15 AM and earliest of June 22 at 3:15 AM. It asks to write cosine and sine equations using this data, sketch the graph, use the equations to predict the Sept 7 sunrise, and find the average sunrise time.

1. The Sunrise Question

2. with solutions…

3. THE QUESTION [info]: The latest sunrise in Montreal was on Dec. 22 at 9:15 AM . According to the almanac, the earliest sunrise occurred on the 22nd of June at 3:15 AM . The sunrise times on other dates can be predicted using a sinusoidal equation. **Assume there is no daylight savings time in Montreal.**

4. THE QUESTION: Part I (a) Write two equations for the described function above; one using cosine and the other using sine.

5. THE SOLUTION: Part I STEPS PROCESS Read the given info and convert it into information that can be used in an equation. For example, the time is not going to be written 9:15 on the graph but 9.25 because 15 min. is a quarter of an hour. Make two lists for the parameters A, B, C & D. One set will be used for the cosine equation and the other will be used for the sine equation. Find the parameters in the mean of DABC [stretches before translations]. To find parameter D, add the min. and max. value and then divide by 2 to find the sinusoidal axis . To find parameter A, subtract the average value from the maximum value to get A. cont‘d on the next slide; Hupsha, hupsha now…

6. STEPS PROCESS cont’d... Parameter B is equal to 2 π divided by the period, which happens to be the number of days in a year; 364. The phase shift (C) is found depending on what kind of equation is being used. If the cosine equation is being found, the maximum value is usually on the y-axis. But the information tells us that the maximum value occurs on Dec. 22, 9 days before Jan. 1 [the y-axis]. The phase shift in the sine equation is determined by finding out the distance of the average value to the y-axis. Finally, to get the equations, plug in the values found into the general formula.

7. THE QUESTION: Part II (b) Sketch the graph for the sinusoidal function described in the problem.

8. THE SOLUTION: Part II STEPS Acknowledge that Dec. 22 is 9 days before Jan. 1. [use this later in graphing stage.] To find the values within the period of one cycle in the graph, subtract nine days from the full, half, and quarter periods. On the y-axis, label where the min. value, max. value and sinusoidal axis are. Plot the points according to info; max value at Dec. 22. Label the axes.

9. THE GRAPH…

10. THE QUESTION: Part III (c) Use one of the equations in (a) to predict the time of sunrise on September 7.

11. THE SOLUTION: Part III Find out what the day of the year September 7 th is by adding up the total number of days in each month up to the given date, assuming there is no daylight savings time. Since d lies along the x-axis, treat the number of days as an x-coordinate and plug in as d in the formula, either sine or cosine and solve. Sun, sun...Mr. Golden Sun.

12. THE QUESTION: Part IV (d) What is the average sunrise time throughout the year?

13. THE SOLUTION: Part IV STEPS PROCESS The avg. sunrise time equals the sinusoidal axis. To find this, find parameter D by adding the minimum and maximum value and divide by two.

14. Looking at the sun gives me a feeling of freedom. Too bad Max can’t feel this way… Sunrise Over Rocks, Lighthouse Beach by flickr user Captain Capture