Hours of Daylight - Anchorage Working with Sinusoidal Models

Transcription

1 Hours of Daylight - Anchorage Working with Sinusoidal Models T he hours of daylight in a given part of the world changes fro day to day. In the northern heisphere, the Suer Solstice (around June 1) is the longest day of the year and the Winter Solstice (around Deceber 1) is the shortest day of the year. The United States Naval Observatory allows website visitors to download the hours of daylight for any day of the year and any location in the world (Source: aa.usno.navy.il). The hours of daylight on the 1 st day of the onth for selected cities are shown in the tables below. Hours of Daylight on 1 st Day of the Month Anchorage, Alaska 06:5 09:39 1:0 15:18 17:59 19: 18:00 15:16 1:1 09:31 06:47 05:7 Hours of Daylight on 1 st Day of the Month Phoenix, Arizona 10:19 11:13 1:10 13:13 14:0 14: 14:0 13:1 1:11 11:10 10:18 09:56 Hours of Daylight on 1 st Day of the Month Easter Island, Chile 13:34 1:5 1:07 11:18 10:41 10:6 10:40 11:17 1:05 1:5 13:34 13:51 Hours of Daylight on 1 st Day of the Month Perth, Australia 14:39 13:5 1:07 10:43 09:37 09:09 09:36 10:4 1:05 13:7 14:40 15:11 1. What is the significance of June 1 for each of the cities? That day is the longest day of the year for Anchorage and Phoenix and is the shortest day of the year for. What is the significance of Deceber 1 for each of the cities? That day is the shortest day of the year for Anchorage and Phoenix and is the longest day of the year for

2 3. On the axes below, draw the scatter plot for the hours of daylight in Anchorage, Alaska. Be sure to convert the ties fro hour-inute forat (e.g. 1:0) to hour forat (e.g. 1.33). 4. Which type of sinusoidal function will best fit the data without requiring a horizontal shift: sin ( ), sin ( ),cos( ),or cos( )? Explain. We first observe that onth 1 (Deceber) is the sae as onth 0 (Deceber of the previous year). Consequently, we predict that the vertical intercept will be 5.45 hours as well. The data appears to start at its iniu value (0, 5.45), clib to it axiu value (6, 19.37), and then descend to its iniu value (1, 5.45). The function h ( ) cos( ) exhibits this sae basic behavior. 5. Find a sinusoidal odel of the type identified in Exercise 4 that fits the data well. The axiu value is and the iniu value is We use these values to find the aplitude and idline. ax + in ax in idline aplitude Since there are 1 onths in a year, we expect that the period of the function is 1 onths. We find B. B period 1 6 h ( ) aplitude ( cos( B ) ) + idline 6.96cos The hours of daylight on the 1 st day of onth are given by h( ) 6.96cos

3 Hours of Daylight - Anchorage Working with Sinusoidal Models T he hours of daylight in a given part of the world changes fro day to day. In the northern heisphere, the Suer Solstice (around June 1) is the longest day of the year and the Winter Solstice (around Deceber 1) is the shortest day of the year. The United States Naval Observatory allows website visitors to download the hours of daylight for any day of the year and any location in the world (Source: aa.usno.navy.il). The hours of daylight on the 1 st day of the onth for selected cities are shown in the tables below. Hours of Daylight on 1 st Day of the Month Anchorage, Alaska 06:5 09:39 1:0 15:18 17:59 19: 18:00 15:16 1:1 09:31 06:47 05:7 Hours of Daylight on 1 st Day of the Month Phoenix, Arizona 10:19 11:13 1:10 13:13 14:0 14: 14:0 13:1 1:11 11:10 10:18 09:56 Hours of Daylight on 1 st Day of the Month Easter Island, Chile 13:34 1:5 1:07 11:18 10:41 10:6 10:40 11:17 1:05 1:5 13:34 13:51 Hours of Daylight on 1 st Day of the Month Perth, Australia 14:39 13:5 1:07 10:43 09:37 09:09 09:36 10:4 1:05 13:7 14:40 15:11 1. What is the significance of June 1 for each of the cities? That day is the longest day of the year for Anchorage and Phoenix and is the shortest day of the year for. What is the significance of Deceber 1 for each of the cities? That day is the shortest day of the year for Anchorage and Phoenix and is the longest day of the year for

4 3. On the axes below, draw the scatter plot for the hours of daylight in Anchorage, Alaska. Be sure to convert the ties fro hour-inute forat (e.g. 1:0) to hour forat (e.g. 1.33). 4. Which type of sinusoidal function will best fit the data without requiring a horizontal shift: sin ( ), sin ( ),cos( ),or cos( )? Explain. We first observe that onth 1 (Deceber) is the sae as onth 0 (Deceber of the previous year). Consequently, we predict that the vertical intercept will be 5.45 hours as well. The data appears to start at its iniu value (0, 5.45), clib to it axiu value (6, 19.37), and then descend to its iniu value (1, 5.45). The function h ( ) cos( ) exhibits this sae basic behavior. 5. Find a sinusoidal odel of the type identified in Exercise 4 that fits the data well. The axiu value is and the iniu value is We use these values to find the aplitude and idline. ax + in ax in idline aplitude Since there are 1 onths in a year, we expect that the period of the function is 1 onths. We find B. B period 1 6 h ( ) aplitude ( cos( B ) ) + idline 6.96cos The hours of daylight on the 1 st day of onth are given by h( ) 6.96cos

5 Worksheet Title Hours of Daylight Anchorage: Working with Sinusoidal Models Filenae: 5031 Keywords NCTM Standard Grade Band Data Type Hours of daylight, sinusoidal odel, odeling, Easter Island, Chile, Naval Observatory, period, aplitude Content Standards Process Standards Nuber and Operations X Proble Solving X Algebra Reasoning and Proof Geoetry Counication Measureent X Connections X Data Analysis and Probability Representations PreK X 9 1 Table License Agreeent At The Make It Real Learning Copany, our goal is to provide quality instructional aterials at a price that even an entry-level teacher can afford. By coplying with this license agreeent, you help us reach that goal. We thank you for your support. Acceptable Use As a paid subscriber, you ay ake hard copies of this worksheet for use in any classes that you teach. This includes traditional teacher-student classes as well as professional developent workshops that you lead. When using the worksheet in a professional developent workshop, this license agreeent ust be included with each copy of the worksheet. Prohibited Use You ay not distribute this worksheet in any for to another person for use in his or her classes. If you are not a paid subscriber, we invite you to subscribe to gain access to a library of worksheets that answer the question, When a I ever going to use this? Subscribe at Thanks.