The Sunrise Question


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The Sunrise Question

  1. 1. The Sunrise Question
  2. 2. with solutions…
  3. 3. THE QUESTION [info]: <ul><li>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. </li></ul><ul><li>**Assume there is no daylight savings time in Montreal.** </li></ul>
  4. 4. THE QUESTION: Part I <ul><li>(a) Write two equations for the described function above; one using cosine and the other using sine. </li></ul>
  5. 5. THE SOLUTION: Part I <ul><li>STEPS </li></ul><ul><li>PROCESS </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>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 . </li></ul><ul><li>To find parameter A, subtract the average value from the maximum value to get A. </li></ul>cont‘d on the next slide; Hupsha, hupsha now…
  6. 6. <ul><li>STEPS </li></ul><ul><li>PROCESS </li></ul><ul><li>cont’d... </li></ul><ul><li>Parameter B is equal to 2 π divided by the period, which happens to be the number of days in a year; 364. </li></ul><ul><li>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]. </li></ul><ul><li>The phase shift in the sine equation is determined by finding out the distance of the average value to the y-axis. </li></ul><ul><li>Finally, to get the equations, plug in the values found into the general formula. </li></ul>
  7. 7. THE QUESTION: Part II <ul><li>(b) Sketch the graph for the sinusoidal function described in the problem. </li></ul>
  8. 8. THE SOLUTION: Part II <ul><li>STEPS </li></ul><ul><li>Acknowledge that Dec. 22 is 9 days before Jan. 1. [use this later in graphing stage.] </li></ul><ul><li>To find the values within the period of one cycle in the graph, subtract nine days from the full, half, and quarter periods. </li></ul><ul><li>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. </li></ul><ul><li>Label the axes. </li></ul>
  9. 9. THE GRAPH…
  10. 10. THE QUESTION: Part III <ul><li>(c) Use one of the equations in (a) to predict the time of sunrise on September 7. </li></ul>
  11. 11. THE SOLUTION: Part III <ul><li>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. </li></ul><ul><li>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. </li></ul><ul><li>Sun, sun...Mr. Golden Sun. </li></ul>
  12. 12. THE QUESTION: Part IV <ul><li>(d) What is the average sunrise time throughout the year? </li></ul>
  13. 13. THE SOLUTION: Part IV <ul><li>STEPS </li></ul><ul><li>PROCESS </li></ul><ul><li>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. </li></ul>
  14. 14. Looking at the sun gives me a feeling of freedom. Too bad Max can’t feel this way… <ul><li>Sunrise Over Rocks, Lighthouse Beach by flickr user Captain Capture </li></ul>