document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=948557>Question 948557</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The distance from the Sun can be approximated by the equation D=2.5cos[0.0172(n-185)+150, where D represents the distance in millions of kilometers and n represents the number of days into a year.\r<BR>\n" );
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document.write( "a) How far will Earth be from the Sun on March 21st, the 80th day of the year? \r<BR>\n" );
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document.write( "This is what I did, but I don't think it's right.<BR>\n" );
document.write( "D=2.5cos[0.0172(80-185)]+150<BR>\n" );
document.write( "=2.5cos(-1.806)+150<BR>\n" );
document.write( "=149.417....\r<BR>\n" );
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document.write( "then I have no clue how to find <BR>\n" );
document.write( "b) The range of distance the Earth can be from the sun.<BR>\n" );
document.write( "c) The period of the cycle, and what it means in context to the question.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #579023 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=579023\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The distance from the Sun can be approximated by the equation D=2.5cos[0.0172(n-185)+150, where D represents the distance in millions of kilometers and n represents the number of days into a year.\r<BR>\n" );
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document.write( "a) How far will Earth be from the Sun on March 21st, the 80th day of the year?\r<BR>\n" );
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document.write( "use the formula to find D.\r<BR>\n" );
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document.write( "Your calculator needs to be set to radians.\r<BR>\n" );
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document.write( "D = 2.5 * cos(0.0172(n-185)+150\r<BR>\n" );
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document.write( "D = distance in millions of kilometers.<BR>\n" );
document.write( "n = the nth day of the year.\r<BR>\n" );
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document.write( "replace n with 80 in that formula and you get:\r<BR>\n" );
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document.write( "D = 2.5 * cos(0.0172 * (80-185)) + 150 \r<BR>\n" );
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document.write( "you got D = 149.4173975 which I believe is a correct approximation based on some other references that I used to check if the answer was reasonable.\r<BR>\n" );
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document.write( "it is.\r<BR>\n" );
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document.write( "b) The range of distance the Earth can be from the sun.\r<BR>\n" );
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document.write( "the range is from 147.5 to 152.5.\r<BR>\n" );
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document.write( "This can be found from the general equation for a cosine trig function.\r<BR>\n" );
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document.write( "that general equation is y = A * cos(B * (x-C)) + D\r<BR>\n" );
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document.write( "A is the amplitude.<BR>\n" );
document.write( "B is the frequency.<BR>\n" );
document.write( "C is the horizontal displacement.<BR>\n" );
document.write( "D is the vertical displacement.\r<BR>\n" );
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document.write( "in your equation, the amplitude is 2.5 and the vertical displacement is 150.\r<BR>\n" );
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document.write( "the vertical displacement is the horizontal axis of the trig function.\r<BR>\n" );
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document.write( "the amplitude is plus or minus from the horizontal axis of the trig function.\r<BR>\n" );
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document.write( "since the axis is at y = 150, then the amplitude is at y = 150 plus or minus 2.5 which makes it y = from 147.5 to y = 152.5.\r<BR>\n" );
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document.write( "c) The period of the cycle, and what it means in context to the question. \r<BR>\n" );
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document.write( "from the basic equation, B is the frequency.\r<BR>\n" );
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document.write( "in your equation B = .0172.\r<BR>\n" );
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document.write( "The frequency is equal to 2*pi / the period.\r<BR>\n" );
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document.write( "the period is equal to 2*pi divided by the frequency.\r<BR>\n" );
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document.write( "your period would therefore be equal to 2*pi / .0172 = 365.3014713\r<BR>\n" );
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document.write( "that's equivalent to the number of days in a year.\r<BR>\n" );
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document.write( "so your period is one year expressed in days.\r<BR>\n" );
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document.write( "the graph of your equation is shown below:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/021801.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "here's a reference that talks about the graph of the sine and cosine equations.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.regentsprep.org/Regents/math/algtrig/ATT7/sinusoidal.htm\" target = \"_blank\">http://www.regentsprep.org/Regents/math/algtrig/ATT7/sinusoidal.htm</a>\r<BR>\n" );
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