Procedure 1. In order to make meaningful conclusions from the data, the sunrise and sunset times are obtained for five different cities, located at approximately 20"N, 30*N, 40N, 50"N, and 60N. Use an atlas to choose cities located at these positions. Of course, if you don't have access to an atlas, you can always use the following cities: 60N-St. Petersburg, Russia 50N-Winnipeg, Canada 40N-Philadelphia, U.S.A. 30N-Cairo, Egypt 20N-Santiago de Cuba, Cuba 2. Sunrise and sunset data and hours of daylight for thousands of locations around the world are readily available at the Time and Date website. Obtain the hours of daylight data for the cities you selected, and complete the following table (use a separate table for each city). The table you find on the Internet reports hours of daylight in "hours:minutes" format, not decimal format Our measurement of time does not use the decimal system (e.g. there are 60 seconds in 1 minute, not 10 or 100) but to graph hours of daylight we must use the decimal system. Therefore, all times in [] format (time) must be converted to [] format (decimal). For example, 13 h and 45 min (13:45) would be 13.75 h in decimal format. City Latitude Longitude Date Day of Year Hours of Daylight, Hours of Daylight, [. Month Day 01 01 0 13:45 13.75 01 15 14 3. Using paper and pencil, construct a scatter plot of the hours of daylight vs the day of the year on the first set of axes (city 1). 4. Visually estimate the amplitude, period, phase shift, and vertical translation of this graph. 5. Using the general form Y = a sin (k(x d)) + C. determine an equation that models this data. Note that the constraints a, d, and c will represent the amplitude, phase shift, and vertical translation, respectively. The constant kis 360" divided by the period. 6. What are the longest and shortest days of the year (t.e., what days receive the most and the least amount of daylight)? Estimate the length of the longest day and the shortest day. What is the range of daylight hours over the course of the year? Which two days receive an equal amount of day-time and night-time? 7. Explain the significance of these four days (the ones you found in step 6). 8. Estimate the daily average number of hours of daylight over the entire year 9. From previous knowledge, it can be easily verified that y = sin(x) passes through the origin. In order to match the hours of daylight function to the general function y=sin(x) where should the origin be placed for translated to)? 10. Estimate the period of the daylight curve. Justify your answer. 11. Using the answers to questions 6 through 10, determine the equation y = a sin (k(x d)) + C. that can be used to model the number of hours of daylight 12. Plot the equation you found in step 11 using Desmos Graphing Calculator. Compare this graph with the graph plotted in step 4. By which method was the best fit achieved? 13. Repeat steps 3 to 12 for the other four cities. 14. Compare and contrast all five equations 15. Compare and contrast all five graphs. For what reasons would the graphs be similar or different 16. Hours of daylight functions can also be modeled using cosine functions. In what ways would this function differ from the sine equation? Determine the cosine equation that could be used to model the data. 2. Find other cities with the same latitude as those in question 2 from the previous page. Would you expect the hours of daylight curves to be similar or different? Confirm your prediction with some research. (8 marks) 3. The longest and shortest days of the year and the days of equal daytime and night-time have special names. What are they called, and why? (5 marks) 4. Can hours of daylight data be modelled as a sinusoidal function for every location on earth? Explain. (3 marks) 5. Find other phenomena that can be modelled using sinusoidal functions. (3 marks) 6. Longitude and latitude are measured in degrees, minutes, and seconds. This is quite similar to our measurement of time. Why is this so? (3 marks) Procedure 1. In order to make meaningful conclusions from the data, the sunrise and sunset times are obtained for five different cities, located at approximately 20"N, 30*N, 40N, 50"N, and 60N. Use an atlas to choose cities located at these positions. Of course, if you don't have access to an atlas, you can always use the following cities: 60N-St. Petersburg, Russia 50N-Winnipeg, Canada 40N-Philadelphia, U.S.A. 30N-Cairo, Egypt 20N-Santiago de Cuba, Cuba 2. Sunrise and sunset data and hours of daylight for thousands of locations around the world are readily available at the Time and Date website. Obtain the hours of daylight data for the cities you selected, and complete the following table (use a separate table for each city). The table you find on the Internet reports hours of daylight in "hours:minutes" format, not decimal format Our measurement of time does not use the decimal system (e.g. there are 60 seconds in 1 minute, not 10 or 100) but to graph hours of daylight we must use the decimal system. Therefore, all times in [] format (time) must be converted to [] format (decimal). For example, 13 h and 45 min (13:45) would be 13.75 h in decimal format. City Latitude Longitude Date Day of Year Hours of Daylight, Hours of Daylight, [. Month Day 01 01 0 13:45 13.75 01 15 14 3. Using paper and pencil, construct a scatter plot of the hours of daylight vs the day of the year on the first set of axes (city 1). 4. Visually estimate the amplitude, period, phase shift, and vertical translation of this graph. 5. Using the general form Y = a sin (k(x d)) + C. determine an equation that models this data. Note that the constraints a, d, and c will represent the amplitude, phase shift, and vertical translation, respectively. The constant kis 360" divided by the period. 6. What are the longest and shortest days of the year (t.e., what days receive the most and the least amount of daylight)? Estimate the length of the longest day and the shortest day. What is the range of daylight hours over the course of the year? Which two days receive an equal amount of day-time and night-time? 7. Explain the significance of these four days (the ones you found in step 6). 8. Estimate the daily average number of hours of daylight over the entire year 9. From previous knowledge, it can be easily verified that y = sin(x) passes through the origin. In order to match the hours of daylight function to the general function y=sin(x) where should the origin be placed for translated to)? 10. Estimate the period of the daylight curve. Justify your answer. 11. Using the answers to questions 6 through 10, determine the equation y = a sin (k(x d)) + C. that can be used to model the number of hours of daylight 12. Plot the equation you found in step 11 using Desmos Graphing Calculator. Compare this graph with the graph plotted in step 4. By which method was the best fit achieved? 13. Repeat steps 3 to 12 for the other four cities. 14. Compare and contrast all five equations 15. Compare and contrast all five graphs. For what reasons would the graphs be similar or different 16. Hours of daylight functions can also be modeled using cosine functions. In what ways would this function differ from the sine equation? Determine the cosine equation that could be used to model the data. 2. Find other cities with the same latitude as those in question 2 from the previous page. Would you expect the hours of daylight curves to be similar or different? Confirm your prediction with some research. (8 marks) 3. The longest and shortest days of the year and the days of equal daytime and night-time have special names. What are they called, and why? (5 marks) 4. Can hours of daylight data be modelled as a sinusoidal function for every location on earth? Explain. (3 marks) 5. Find other phenomena that can be modelled using sinusoidal functions. (3 marks) 6. Longitude and latitude are measured in degrees, minutes, and seconds. This is quite similar to our measurement of time. Why is this so