470 CHAPTER 6 Trigonometric Functions Chapter Project Length of Day Revisited Go to http://en.wikipedia.org/wiki/latitude and read about latitude. Then go to http://www.orchidculture.com/COD/daylength.html 1. For a particular latitude, record in a table the length of day for the various days of the year. For January 1, use 1 as the day, for January 16, use 16 as the day, for February 1, use 32 as the day, and so on. Enter the data into an Excel spreadsheet using column B for the day of the year and column C for the length of day. 2. Draw a scatter plot of the data with day of the year as the independent variable and length of day as the dependent variable using Excel. (The Chapter 2 project describes how to draw a scatter plot in Excel.) 3. Determine the sinusoidal function of best fit, ( ) = + + y A Bx C D sin , as follows: (a) Enter initial guesses for the values of A, B, C, and D into column A with the value of A in cell A1, B in cell A2, C in cell A3, and D in cell A4. (b) Enter ( ) = + + “ A$1*sin A$2*B1 A$3 A$4” into cell D1. Copy this cell entry into the cells below D1 for as many rows as there are data. For example, if column C goes to row 23, then column D should also go to row 23. (c) Enter ( ) = − “ D1 C1 ^2” into cell E1. Copy this entry below cell E1 as described in part 3(b). (d) The idea behind curve fitting is to make the sum of the squared differences between what is predicted and actual observations as small as possible. Enter ( ) =“ sum E1..E# ” into cell A6, where # represents the row number of the last data point. For example, if you have 23 rows of data, enter “ ( ) =sum E1..E23 ” in cell A6. (e) Now, install the Solver feature of Excel. To do this, click the File tab, and then select Options. Select Add-Ins. In the drop-down menu entitled “Manage,” choose Excel Addins, and then click Go . . . . Check the box entitled “Solver Add-in” and click OK. The Solver add-in is now available in the Data tab. Choose Solver. Fill in the screen as shown on the next page. Click Solve. The values for A, B, C, and D are located in cells A1–A4. What is the sinusoidal function of best fit? Internet-based Project Credit: Skylines/Shutterstock Citation: Excel © 2022 Microsoft Corporation. Used with permission from Microsoft. 4. Determine the longest day of the year according to your model. What is the day length on the longest day of the year? Determine the shortest day of the year according to your model.What is the day length on the shortest day of the year? 5. On which days is the day length exactly 12 hours according to your model? 6. Look up the day on which the vernal equinox and autumnal equinox occur. How do they match up with the results obtained in part 5? 7. Do you think your model accurately describes the relation between day of the year and length of day? 8. Use your model to predict the hours of daylight for the latitude you selected for various days of the year. Go to the Old Farmer’s Almanac or other website (such as http://astro.unl.edu/ classaction/animations/coordsmotion/daylighthoursexplorer. html) to determine the hours of daylight for the latitude you selected. How do the two compare?
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