document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206910>Question 1206910</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.\r<BR>\n" );
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document.write( "y = − 1/2 cos (6𝜋x/5 + 3)-7\r<BR>\n" );
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document.write( "2. A Ferris wheel has a radius of 16 m and its centre is 18 m above the ground. It rotates once every 60 s. Ethan gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. Determine both a sine and cosine function that gives the height, ℎ, of point P above the ground at any time,𝑡𝑡, where ℎ is in metres and 𝑡𝑡 is in seconds. Both functions must have the smallest possible phase shift. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #844657 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=844657\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Determine the period, amplitude, vertical displacement, and phase shift for the following function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-+%281%2F2%29+cos+%28%286pi%2F5%29%2Ax+%2B+3%29-7 ALIGN=MIDDLE  ALT=\"y+=+-+%281%2F2%29+cos+%28%286pi%2F5%29%2Ax+%2B+3%29-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The general form of the cosine function is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+a+%2Acos%28b%2Ax%2Bc%29+%2B+d ALIGN=MIDDLE  ALT=\"y+=+a+%2Acos%28b%2Ax%2Bc%29+%2B+d\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where:\r<BR>\n" );
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document.write( "amplitude = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28a%29 ALIGN=MIDDLE  ALT=\"abs%28a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => amount of travel above and below midpoint\r<BR>\n" );
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document.write( "period = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2Fabs%28b%29 ALIGN=MIDDLE  ALT=\"2pi%2Fabs%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > => time to complete one cycle\r<BR>\n" );
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document.write( "phase shift = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-c%2Fb ALIGN=MIDDLE  ALT=\"-c%2Fb\"   DISABLED_style= \"max-width:90%; height:auto;\" > => how far from zero the cycle starts\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > = vertical shift of the function\r<BR>\n" );
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document.write( "in your case\r<BR>\n" );
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document.write( "the period: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F%286pi%2F5%29=+5%2F3 ALIGN=MIDDLE  ALT=\"2pi%2F%286pi%2F5%29=+5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "amplitude: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28-1%2F2%29=1%2F2 ALIGN=MIDDLE  ALT=\"abs%28-1%2F2%29=1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "vertical displacement: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-7 ALIGN=MIDDLE  ALT=\"d=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "phase shift :  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-%28c%2Fb%29+=+-3%2F%286pi%2F5%29=-1%2F%282pi%2F5%29=-5%2F2pi ALIGN=MIDDLE  ALT=\"-%28c%2Fb%29+=+-3%2F%286pi%2F5%29=-1%2F%282pi%2F5%29=-5%2F2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "2. A Ferris wheel has a radius of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16m ALIGN=MIDDLE  ALT=\"16m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and its center is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18m ALIGN=MIDDLE  ALT=\"18m\"   DISABLED_style= \"max-width:90%; height:auto;\" > above the ground. It rotates once every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60s ALIGN=MIDDLE  ALT=\"60s\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Ethan gets on the Ferris wheel at its lowest point and then the wheel starts to rotate. \r<BR>\n" );
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document.write( "Determine both a sine and cosine function that gives the height, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" >, of point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P ALIGN=MIDDLE  ALT=\"P\"   DISABLED_style= \"max-width:90%; height:auto;\" > above the ground at any time, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h ALIGN=MIDDLE  ALT=\"h\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in meters and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+ ALIGN=MIDDLE  ALT=\"t+\"   DISABLED_style= \"max-width:90%; height:auto;\" >is in seconds. Both functions must have the smallest possible phase shift.\r<BR>\n" );
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document.write( "Since the Ferris wheel starts with the rider at the bottom (minimum point in the cycle), we can represent the rider’s position with a <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=negative ALIGN=MIDDLE  ALT=\"negative\"   DISABLED_style= \"max-width:90%; height:auto;\" > cosine function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+-a+%2Acos%28bt%2Bc%29+%2B+d ALIGN=MIDDLE  ALT=\"h%28t%29+=+-a+%2Acos%28bt%2Bc%29+%2B+d\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "given:\r<BR>\n" );
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document.write( " amplitude is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=abs%2816%29 ALIGN=MIDDLE  ALT=\"a=abs%2816%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Since we have a negative cosine function, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+=+-16 ALIGN=MIDDLE  ALT=\"a+=+-16\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "It rotates once every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=60s ALIGN=MIDDLE  ALT=\"60s\"   DISABLED_style= \"max-width:90%; height:auto;\" >=> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi%2F60=pi%2F30 ALIGN=MIDDLE  ALT=\"2pi%2F60=pi%2F30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "The rider starts at the bottom, so there is no phase shift, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=0 ALIGN=MIDDLE  ALT=\"c=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We are given that the center of the wheel is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18m ALIGN=MIDDLE  ALT=\"18m\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the ground, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=18 ALIGN=MIDDLE  ALT=\"d=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+-16cos%28%28pi%2F30%29t%2B0%29+%2B+18 ALIGN=MIDDLE  ALT=\"h%28t%29+=+-16cos%28%28pi%2F30%29t%2B0%29+%2B+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29+=+-16cos%28%28pi%2F30%29t%29+%2B+18 ALIGN=MIDDLE  ALT=\"h%28t%29+=+-16cos%28%28pi%2F30%29t%29+%2B+18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "or sine function:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=16sin%28%28pi%2F30%29t%29%2B18 ALIGN=MIDDLE  ALT=\"h%28t%29=16sin%28%28pi%2F30%29t%29%2B18\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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