document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=934401>Question 934401</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A paddle wheel has a 10ft. diameter.  At t=0, Jenn notices a weed stuck to the top of the paddle, which is 8ft above the surface of the water.  It takes 4 seconds for the paddle to return to the same position.  Find a function that will model the height of the weed above the water at any time (t).<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #567823 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=567823\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your equation is going to be:\r<BR>\n" );
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document.write( "y = 5cos(90x)+3\r<BR>\n" );
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document.write( "your calculator, if you have one that graphs, should be set to degree mode.\r<BR>\n" );
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document.write( "the basic equation that is used 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( "c is equal to 0 because there is no horizontal displacement.\r<BR>\n" );
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document.write( "if a is equal to 1 and b is equal to 1 and d is equal to 0 and c is equal to 0, the equation becomes:\r<BR>\n" );
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document.write( "y = cos(x).\r<BR>\n" );
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document.write( "the cos(x) function is equal to 1 when x is equal to 0 degrees and is equal to 0 when x is equal to 180 degrees\r<BR>\n" );
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document.write( "that function looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/122001.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "the period is equal to 360 degrees.\r<BR>\n" );
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document.write( "we want the period to be equal to 4 degrees in order to simulate 4 seconds where each degree is equal to 1 second.\r<BR>\n" );
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document.write( "the formula for frequency is that frequency = 360 / period.\r<BR>\n" );
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document.write( "the formula for period is that period = 360 / frequency.\r<BR>\n" );
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document.write( "we want to find the frequency, so we use the frequency formula.\r<BR>\n" );
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document.write( "we get frequency = 360 / 4 which is equal to 90.\r<BR>\n" );
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document.write( "if we make frequency equal to 90, than the period of the cosine function, where it goes through one complete cycle, will be 4.\r<BR>\n" );
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document.write( "our formula becomes:\r<BR>\n" );
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document.write( "y = cos(90x) with nothing else changed.\r<BR>\n" );
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document.write( "the graph of that equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/122002.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "that looks a lot like the last graph only the period is equal to 4 rather than 360.\r<BR>\n" );
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document.write( "the diameter of the paddle wheel is equal to 10 which means the radius is equal to 5.\r<BR>\n" );
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document.write( "we want the graph of the paddle wheel to have a high point of 5 and a low point of 5 rather than a high point of 1 and a low point of 1.\r<BR>\n" );
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document.write( "we make our a factor in the equation of y = a * cos(b * (x-c)) + d equal to 5 and keep our b factor to 90 and our equation becomes:\r<BR>\n" );
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document.write( "y = 5 * cos(90 * x)\r<BR>\n" );
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document.write( "the graph of that equation looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/122003.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "we're almost there.\r<BR>\n" );
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document.write( "we want the height of the weed that is stuck to the paddle to be 8 feet high above the water line when the number of seconds is 0.\r<BR>\n" );
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document.write( "we have to add 3 to the height of out graph because 5 + 3 will be equal to 8.\r<BR>\n" );
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document.write( "we do this by changing d from 0 to 3.\r<BR>\n" );
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document.write( "our equation now becomes:\r<BR>\n" );
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document.write( "y = 5 * cos(90 * x) + 3\r<BR>\n" );
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document.write( "that graph looks like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2014/122004.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "x represents the seconds.\r<BR>\n" );
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document.write( "the center of the paddle wheel is at y = 3\r<BR>\n" );
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document.write( "the radius of the paddle wheel is equal to 5.\r<BR>\n" );
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document.write( "the water line is at y = 0\r<BR>\n" );
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document.write( "when x = 0, y = 8 feet high above the water line which is what we want.\r<BR>\n" );
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document.write( "when x = 4, the cycle completes and the height of the weed is 8 again.\r<BR>\n" );
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document.write( "in between are the different intermediate cycles of the function of y = 5 * cos(90 * x) + 3\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "0 seconds = 8<BR>\n" );
document.write( "1 second = 3<BR>\n" );
document.write( "2 seconds = -2<BR>\n" );
document.write( "3 seconds = 3<BR>\n" );
document.write( "4 seconds = 8\r<BR>\n" );
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document.write( "y = 3 is the center line of the cosine function.  <BR>\n" );
document.write( "it fluctuates plus 5 from that and - 5 from that.\r<BR>\n" );
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document.write( "it is also the center of the paddle wheel.<BR>\n" );
document.write( "it's the hub that the paddle wheel revolves around.\r<BR>\n" );
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document.write( "y = 0 is the water line.<BR>\n" );
document.write( "anything above that is in the air above the water.<BR>\n" );
document.write( "anything below that is in the water.\r<BR>\n" );
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document.write( "the weed stuck on the paddle wheel goes round and round every 4 seconds.<BR>\n" );
document.write( "it's high point is 8 feet above the water line.<BR>\n" );
document.write( "it's low  point is 2 feet below the water line.\r<BR>\n" );
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document.write( "using this equation, we can find the height of the weed any number of seconds into the revolution of the paddle wheel.\r<BR>\n" );
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document.write( "simply replace x in the equation and solve for y using your calculator in degree mode.\r<BR>\n" );
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document.write( "the equation is, once again, y = 5 * cos(90 * x) + 3<BR>\n" );
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