Answer Happy

In order to create a function from this, we need to quantify the date into "day of year", beginning with January 1, which we will call "Day 1" (we can assume 1 corresponds to 11:59 PM on January 1, so we'll think of the New Year beginning at x-0). This becomes our independent variable. We also turn the amount of time between sunrise and sunset into "hours of daylight", which becomes our dependent variable. Date Day of Sunrise Sunset Minutes of Hours of Year Daylight Daylight 1/3 3 7:20 4:43 563 9.38 1/10 10 7:20 4:49 569 9.48 1/17 17 7:17 4:57 580 9.67 1/24 24 7:13 5:05 592 9.87 1/31 31 7:08 5:14 606 10.1 2/7 38 7:01 5:22 621 10.35 2/14 45 6:52 5:3 639 10.65 2/21 52 6:43 5:39 656 10.93 2/28 59 6:33 5:47 674 11.23 3/7 66 6:22 5:55 693 11.55 3/14 73 6:11 6:02 711 11.85 3/21 80 6:00 6:10 730 12.17 3/28 87 5:48 6:17 749 12.48 4/4 94 5:37 6:24 767 12.78 4/11 101 5:25 6:31 786 13.1 4/18 108 5:15 6:38 803 13.38 4/25 115 5:05 6:46 821 13.68 5/2 122 4:56 6:53 837 13.95 5/9 129 4:48 7:00 852 14.2 5/16 136 4:41 7:07 866 14.43 5/23 143 4:35 7:13 878 14.63 5/30 150 4:31 7:19 888 14.8 6/6 157 4:28 7:24 896 14.93 6/13 164 4:27 7:28 901 15.02 6/20 171 4:27 7:30 903 15.05 6/27 178 4:29 7:31 902 15.03 7/4 185 4:32 7:31 899 14.98 192 7/11 4:37 7:28 14.85 891 199 4:42 7/18 7:25 883 14.72 206 7/25 4:48 7:19 14.52 871 8/1 213. 4:55 7:13 858 14.3 8/8 220 5:01 7:05 844 14.07 8/15 5:08 227 6:56 13.8 828 234 8/22 5:15 6:46 13.52 811 241 8/29 5:22 6:35 793 13.22 9/5 248 5:28 6:24 776 12.93
9/12 255 5:35 6:13 758 12.63 9/19 262 5:42 6:01 739 12.32 9/26 269 5:48 5:49 721 12.02 10/3 276 5:55 5:37 702 11.7 10/10 283 6:02 5:26 684 11.4 10/17 290 6:10 5:15 665 11.08 10/24 297 6:17 5:05 648 10.8 10/31 304 6:25 4:56 631 10.52 11/7 311 6:34 4:48 614 10.23 11/14 318 6:42 4:42 600 10 11/22 326 6:50 4:34 588 9.8 11/29 333 6:58 4:31 573 9.55 12/6 340 7:05 4:30 565 9.42 12/13 347 7:12 4:32 560 9.33 12/20 354 7:16 4:34 558 9.3 12/27 361 7:19 4:37 558 9.3 1. The sinusoidal function that can be used to model this data is f(x) = 2.836 sin (0.0172(x-80))+12.164, where x is the day of the year, and f(x) gives the number of hours of daylight on day x.
1. The sinusoidal function that can be used to model this data is f(x)=2.836 sin (0.0172(x-80))+12.164, where x is the day of the year, and f(x) gives the number of hours of daylight on day x. a. What are the amplitude, period, and midline of the function? Round to the nearest thousandth. b. Do these values make sense based on the data? Why or why not? 2. Since this is a sinusoidal function, the domain is all real numbers. However, this function has a practical application. What is the practical domain of this function? Why? 3. Evaluate the model for July 4 and October 31. How close is the model to the actual data? Round to the nearest hundredth. 4. Find the first and second derivatives of the model without using technology. Show and label all your work. 5. a. Find the critical values of the function. Show and label your work. Round to the nearest hundredth. b. Use the first or second derivative tests to determine the maximum and minimum values of the function, and on which dates they occur. Show and label all your written work. Round to the nearest hundredth. C. Do the dates make sense based on what you know about the seasons?
1. The sinusoidal function that can be used to model this data is f(x)=2.836 sin (0.0172(x-80))+12.164, where x is the day of the year, and f(x) gives the number of hours of daylight on day x. a. What are the amplitude, period, and midline of the function? Round to the nearest thousandth. b. Do these values make sense based on the data? Why or why not? 2. Since this is a sinusoidal function, the domain is all real numbers. However, this function has a practical application. What is the practical domain of this function? Why? 3. Evaluate the model for July 4 and October 31. How close is the model to the actual data? Round to the nearest hundredth. 4. Find the first and second derivatives of the model without using technology. Show and label all your work. 5. a. Find the critical values of the function. Show and label your work. Round to the nearest hundredth. b. Use the first or second derivative tests to determine the maximum and minimum values of the function, and on which dates they occur. Show and label all your written work. Round to the nearest hundredth. C. Do the dates make sense based on what you know about the seasons?