document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174072>Question 1174072</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small lake on the island contains two species of fish; hake and redfish. The hake are predators that eat the redfish. The fish population in the lake varies periodically with period 180 days. The number of hake varies between 500 and 1500, and the number of redfish varies between 1000 and 3000. The hake reach their maximum population 30 days after the redfish have reached their maximum population in the cycle. Find cosine functions of the form y=a cos k(t-b) + c that model the hake and redfish populations in the lake. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #799455 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=799455\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the formula that i have is:\r<BR>\n" );
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document.write( "y = a * cos( b * (x - c) ) + d\r<BR>\n" );
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document.write( "i'll use that formula because i'm more familiar with it and the references that i have use the same terminology.\r<BR>\n" );
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document.write( "in that formula:\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( "x is the angle in degrees or radians, whichever you choose.  i choose degrees for this problem.<BR>\n" );
document.write( "c is the horizontal shift<BR>\n" );
document.write( "d is the vertical shift.\r<BR>\n" );
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document.write( "your formula is:\r<BR>\n" );
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document.write( "y = a * cos( k * (t - b) ) + c\r<BR>\n" );
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document.write( "your a is equal to my a.<BR>\n" );
document.write( "your k is equal to my b.<BR>\n" );
document.write( "your t is equal to my x.<BR>\n" );
document.write( "your b is equal to my c.<BR>\n" );
document.write( "your c is equal to my d.\r<BR>\n" );
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document.write( "here they are, one on top of the other, so you can see the relationship visually.\r<BR>\n" );
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document.write( "mine:<BR>\n" );
document.write( "y = a * cos( b * (x - c) ) + d<BR>\n" );
document.write( "yours:<BR>\n" );
document.write( "y = a * cos( k * (t - b) ) + c\r<BR>\n" );
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document.write( "when all is done, my formula becomes:\r<BR>\n" );
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document.write( "for the hake:<BR>\n" );
document.write( "y = 500 * cos( 2 * (x - 30)) ) + 1000<BR>\n" );
document.write( "for the redfish:<BR>\n" );
document.write( "y = 1000 * cos( 2 * x) + 2000\r<BR>\n" );
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document.write( "the graph of both equations is shown below:\r<BR>\n" );
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document.write( "the red line is the graph of the redfish equation.<BR>\n" );
document.write( "the blue line is the graph of the hake equation.\r<BR>\n" );
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document.write( "why the equations are what they are follows:\r<BR>\n" );
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document.write( "the length of the cycle for both fish is 180 days.<BR>\n" );
document.write( "to model this, you want the length of the cycle of the cosine waves to be 180 degrees.<BR>\n" );
document.write( "in this way, one degree is the same as one day.<BR>\n" );
document.write( "the formula to use if:<BR>\n" );
document.write( "frequency = 360 / period<BR>\n" );
document.write( "from this formula, you also get:<BR>\n" );
document.write( "period = 360 / frequency<BR>\n" );
document.write( "since you want the period ti be 180 days, then formula for frequency becomes:<BR>\n" );
document.write( "frequency = 360 / 180 = 2.\r<BR>\n" );
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document.write( "the number of hake varies between 1500 and 500.<BR>\n" );
document.write( "that puts the center line of the hake equation halfway between these at 1000.<BR>\n" );
document.write( "the vertical shift for the hake is therefore 1000.\r<BR>\n" );
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document.write( "the number of redfish varies between 1000 and 3000.<BR>\n" );
document.write( "that puts the center line of the redfish equation halfway between these at 2000.<BR>\n" );
document.write( "the vertical shift for the redfish is therefore 2000.\r<BR>\n" );
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document.write( "the hake population peaks 30 days after the redfish population peaks.<BR>\n" );
document.write( "this means that the equation for the hake has a horizontal shift of 30 days.<BR>\n" );
document.write( "this is why the redfish equation contains 2 * x while the hake equation contains 2 * (x - 30).<BR>\n" );
document.write( "there is no horizontal shift for the redfish while there is a 30 day horizontal shift for the hake.<BR>\n" );
document.write( "the effect is to shift the hake equation 30 days to the right which makes it peak 30 days later than the redfish.\r<BR>\n" );
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document.write( "that pretty much does it.<BR>\n" );
document.write( "i hope i explained it well.<BR>\n" );
document.write( "if you have any further questions about this, feel free to ask.\r<BR>\n" );
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document.write( "here's a reference that might help.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://mathbitsnotebook.com/Algebra2/TrigGraphs/TGoutline.html\" target = \"_blank\">https://mathbitsnotebook.com/Algebra2/TrigGraphs/TGoutline.html</a>\r<BR>\n" );
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