document.write( "Question 40525: the sun rises at 09:17 on dec. 21 and at 04:35 on June 22> there is no daylight savings time, the time the sun rises on any other date can be predicted from a sinusoidal grpah with period of 365 days.
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Algebra.Com's Answer #25942 by AnlytcPhil(1806)\"\" \"About 
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the sun rises at 09:17 on dec. 21 and at 04:35 on June 22> there is no daylight\r\n" );
document.write( "savings time, the time the sun rises on any other date can be predicted from a\r\n" );
document.write( "sinusoidal graph with period of 365 days.\r\n" );
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document.write( "Write a sinusoidal equation that relates the time the sun rises to the day of\r\n" );
document.write( "the year\r\n" );
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document.write( "This is obviously referring to some location on earth nearer the North pole\r\n" );
document.write( "than the continental US.  Does that perhaps happen in northern Alaska?  \r\n" );
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document.write( "Anyway,\r\n" );
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document.write( "Let S(t) represent the time of dawn on day number t.\r\n" );
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document.write( "To find the number of day of the year that June 22 is, we add up the numbers\r\n" );
document.write( "of days of the months up through May\r\n" );
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document.write( "Jan has 31\r\n" );
document.write( "Feb has 28\r\n" );
document.write( "Mar has 31\r\n" );
document.write( "Apr has 30\r\n" );
document.write( "May has 31\r\n" );
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document.write( "       151\r\n" );
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document.write( "and then add another 22 to 151 and get that June 22 is the 173rd of the \r\n" );
document.write( "year.\r\n" );
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document.write( "Since December has 31 days, December 21 is 10 days before the end of the\r\n" );
document.write( "year, so we subtract 10 from 365 and get that December 21 is the 355th \r\n" );
document.write( "day of the year. \r\n" );
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document.write( "4:35 AM is 4 + 35/60 or 55/12 hours. \r\n" );
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document.write( "So when t = 173, S(t) = 55/12 \r\n" );
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document.write( "9:17 AM is 9 + 17/60 or 557/60 hours.\r\n" );
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document.write( "So when t = 355, S(t) = 557/60\r\n" );
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document.write( "Use the equation \r\n" );
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document.write( "S(t) = A sin(Bt + C) + D\r\n" );
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document.write( "The maximum value of the sine function is 1 which occurs when the angle\r\n" );
document.write( "is p/2, so when t = 355, Bt + C = p/2\r\n" );
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document.write( "B·(355) + C = p/2\r\n" );
document.write( "355B + C = p/2\r\n" );
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document.write( "The minimum value of the sine function is -1 which occurs when the angle\r\n" );
document.write( "is 3p/2, so when t = 173, Bt + C = 3p/2\r\n" );
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document.write( "B·(173) + C = p/2\r\n" );
document.write( "173B + C = p/2\r\n" );
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document.write( "So we solve the system of equations\r\n" );
document.write( "(Best to use the MATRIX operations on a graphical calculator) \r\n" );
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document.write( "355B + C =  p/2\r\n" );
document.write( "173B + C = 3p/2\r\n" );
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document.write( "and get\r\n" );
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document.write( "B = -.0172514981\r\n" );
document.write( "C = 7.698628151\r\n" );
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document.write( "It is convenient to use a graphical caluculator and store\r\n" );
document.write( "those two values respectively as B and C.\r\n" );
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document.write( "so    \r\n" );
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document.write( "Now when t = 355, S(t) = 557/60\r\n" );
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document.write( "So we plug that into\r\n" );
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document.write( "S(t) = A sin(Bt + C) + D\r\n" );
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document.write( "and get\r\n" );
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document.write( "S(355) = 557/60 = A sin(355B + C) + D\r\n" );
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document.write( "Also when t = 173, S(t) = 55/12\r\n" );
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document.write( "So we plug that into\r\n" );
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document.write( "S(t) = 55/12 = A sin(173B + C) + D\r\n" );
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document.write( "Now we have this system of two equations in two unknowns:\r\n" );
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document.write( "A sin(355B + C) + D = 557/60 \r\n" );
document.write( "A sin(173B + C) + D = 55/12\r\n" );
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document.write( "Using the matrix utility of a graphing calculator with\r\n" );
document.write( "the stored values of B and C, and we get\r\n" );
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document.write( "A = 2.35, D = 6.9333333333\r\n" );
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document.write( "So the equation is\r\n" );
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document.write( "S(t) = A sin(Bt + C) + D\r\n" );
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document.write( "with A = 2.35, B = -.0172614981, C = 7.698628151, D = 6.933333333\r\n" );
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document.write( "Edwin\r\n" );
document.write( "AnlytcPhil@aol.com
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