document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1142011>Question 1142011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 12.	The nearby beach has an average depth of the water at low tide of 1 m at midnight. The average depth of the water at high tide is 8 m at 6am. One complete cycle takes 12 h.<BR>\n" );
document.write( "(a) Determine the equation that models the tides using cosine as the base function. <BR>\n" );
document.write( "(b) Many people dive from this pier during the day.  If the water must be at least 3 m deep to dive safely, between what daylight hours should people dive? <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 #762673 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=762673\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation will be:\r<BR>\n" );
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document.write( "y = 3.5 * cos(30 * (x + 6)) + 4.5\r<BR>\n" );
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document.write( "it looks like this.\r<BR>\n" );
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document.write( "the general form of the cosine wave 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( "a is the amplitude<BR>\n" );
document.write( "b is the frequency<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 amplitude is 3.5.\r<BR>\n" );
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document.write( "that's because your cosine wave will go from 8 to 1, which is a difference of 7.\r<BR>\n" );
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document.write( "divide that 2 and you get your amplitude of 3.5\r<BR>\n" );
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document.write( "that would be \"a\" in the general equation.\r<BR>\n" );
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document.write( "since your low tide is 1 and your high tide is 8, then your cosine wave has to go from 8 to 1.\r<BR>\n" );
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document.write( "that would put the center line at 4.5 because there are 3.5 units between 8 and 4.5 and there are 3.5 units between 4.5 and 1.\r<BR>\n" );
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document.write( "since your center line is at y = 4.5, then your vertical shift is 4.5.\r<BR>\n" );
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document.write( "that would be \"d\" in the general equation.\r<BR>\n" );
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document.write( "your period if the interval for one full wave.\r<BR>\n" );
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document.write( "since the wave repeats every 12 hours, then your period is equal to 12.\r<BR>\n" );
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document.write( "your frequency is equal to 360 degrees / 12 = 30.\r<BR>\n" );
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document.write( "that would be \"b\" in the general equation.\r<BR>\n" );
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document.write( "so far your cosine wave is 3.5 * cos(30 * x) + 4.5\r<BR>\n" );
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document.write( "your cosine wave normally starts at its high point.\r<BR>\n" );
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document.write( "that would be y = 8 in the equation as shown.\r<BR>\n" );
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document.write( "you want it to be at 1.\r<BR>\n" );
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document.write( "you can do one of two things.\r<BR>\n" );
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document.write( "if you make the amplitude negative, then the cosine wave will start at 1 rather than 8.\r<BR>\n" );
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document.write( "you can also do a horizontal shift.\r<BR>\n" );
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document.write( "since the cosine is 1 when x = 6, you need to shift it to the left 6 units so it will be y = 1 when x = 0.\r<BR>\n" );
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document.write( "that would be c in the general form of the equation.\r<BR>\n" );
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document.write( "that gets you the final equation of y = 3.5 * cos(30 * (x + 6)) + 4.5\r<BR>\n" );
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document.write( "that's what's on the graph above.\r<BR>\n" );
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document.write( "if the water needs to be at least 3 feet deep, then draw a line at y = 3 and check the times when the graph is at or above that line.\r<BR>\n" );
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document.write( "those times would be between x = 2.154 and 9.846, and between x = 14.154 and 21.846.\r<BR>\n" );
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document.write( "those hours correspond to military time which goes from 0 to 24.\r<BR>\n" );
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document.write( "x = 2.154 corresponds to somewhere between 2:14 and 2:15 am.<BR>\n" );
document.write( "x = 9.846 corresponds to somewhere between 9:50 and 9:51 am.<BR>\n" );
document.write( "x = 14.154 corresponds to somewhere between 2:14 and 2:15 pm.<BR>\n" );
document.write( "x = 21.846 corresponds to somewhere between 9:50 and 9:51 pm.\r<BR>\n" );
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document.write( "assuming daylight is from 6:00 am to 8:00 pm, then you would get the following.\r<BR>\n" );
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document.write( "water is at least 3 feet deep between the hours of 6:00 am and 9:51 am.<BR>\n" );
document.write( "water is at least 3 feet deep between the hours of 2.14 pm and 8:00 pm.\r<BR>\n" );
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document.write( "since it was not specified when the daylight hours start and stop, i assumed between 6:00 am and 8:00 pm.\r<BR>\n" );
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document.write( "you can adjust based on what you know.\r<BR>\n" );
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