document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1069701>Question 1069701</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose that the height h(t) (in meters) of a cabin of the London Eye as a function of the time t (in minutes) can be written as \r<BR>\n" );
document.write( "\n" );
document.write( "                 h(t)=a + b * cos(c * t)\r<BR>\n" );
document.write( "\n" );
document.write( "with a, b and c constants. \r<BR>\n" );
document.write( "\n" );
document.write( "At time t=0 the cabin should start at the bottom. Use the data above to find the coefficients a, b and c. For c, work in radians. Give exact answers.\r<BR>\n" );
document.write( "\n" );
document.write( "Below is the answer provided with my questions <<<  INTERSPERSED  >>>: \r<BR>\n" );
document.write( "\n" );
document.write( "<<< I CAN'T VISUALIZE HOW COS COMES INTO THIS -- BY CONTRAST I THINK I COULD SEE USING SIN CAUSE = OPP/HYP, SO MULTIPLYING IT BY RADIUS (WHICH IS HYP) WOULD GIVE OPP, WHICH WOULD BE THE Y-COMPONENT OR THE DIFFERENCE OR ADDITION IN HEIGHT FROM OR TO THE HEIGHT OF THE AXIS>>>\r<BR>\n" );
document.write( "\n" );
document.write( "It takes 30 minutes for a complete tour. This implies that 30* c = 2 * pi; \r<BR>\n" );
document.write( "\n" );
document.write( "     <<<  IN 30 MIN. THEY TRAVEL ONE REVOLUTION OR 2 * PI. OK. BUT WHY DOES THIS MEAN THAT THE VALUE IN THE COS PARENTHESES HAS TO EQUAL THAT 2 * PI? <BR>\n" );
document.write( "AND WHAT HAPPENED TO A & B ... WHAT IS THIS THOUGHT PROCESS THAT TAKES THE COS BY ITSELF?  >>>\r<BR>\n" );
document.write( "\n" );
document.write( "and therefore c= pi / 15. \r<BR>\n" );
document.write( "\n" );
document.write( "So we have h(t)=a+b*cos((pi/15) * t).\r<BR>\n" );
document.write( "\n" );
document.write( "Further we have\r<BR>\n" );
document.write( "\n" );
document.write( "h(0)=0:a+b=0  \r<BR>\n" );
document.write( "\n" );
document.write( "and  h(15)=135: a-b = 135.\r<BR>\n" );
document.write( "\n" );
document.write( "This implies that\r<BR>\n" );
document.write( "\n" );
document.write( "a = -b = 135/2 <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #684855 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=684855\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I read that the London Eye is 135 m tall and has a diameter of 120 m.<BR>\n" );
document.write( "That tells me that the bottom is at a height of 15 m.<BR>\n" );
document.write( "We need to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c ALIGN=MIDDLE  ALT=\"c\"   DISABLED_style= \"max-width:90%; height:auto;\" > to make<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=a+%2B+b+%2A+cos%28c+%2A+t%29 ALIGN=MIDDLE  ALT=\"h%28t%29=a+%2B+b+%2A+cos%28c+%2A+t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > a good model for your height above the ground<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes after you start at the bottom.<BR>\n" );
document.write( " <BR>\n" );
document.write( "WHY COSINE:<BR>\n" );
document.write( "We would like <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be the start of the ride,<BR>\n" );
document.write( "where you probably are at the bottom at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h=15m ALIGN=MIDDLE  ALT=\"h=15m\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and cosine maxes out at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cosine%28c%2A0%29=cos%280%29=1 ALIGN=MIDDLE  ALT=\"cosine%28c%2A0%29=cos%280%29=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That works for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=a+%2B+b+%2A+cos%28c+%2A+t%29 ALIGN=MIDDLE  ALT=\"h%28t%29=a+%2B+b+%2A+cos%28c+%2A+t%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3C0 ALIGN=MIDDLE  ALT=\"b%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%280%29=a%2Bb%2Acos%280%29=a%2Bb=15m ALIGN=MIDDLE  ALT=\"h%280%29=a%2Bb%2Acos%280%29=a%2Bb=15m\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "You could use sine, but <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sine%280%29=0 ALIGN=MIDDLE  ALT=\"sine%280%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "would not make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > the bottom height.<BR>\n" );
document.write( "It would be the middle of the ride.\r<BR>\n" );
document.write( "\n" );
document.write( " <BR>\n" );
document.write( "THE AMPLITUDE:<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3C0 ALIGN=MIDDLE  ALT=\"b%3C0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the amplitude of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%2Acos%28c%2At%29 ALIGN=MIDDLE  ALT=\"b%2Acos%28c%2At%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28b%29 ALIGN=MIDDLE  ALT=\"abs%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%3C=b%2Acos%28c%2At%29%3C=-b ALIGN=MIDDLE  ALT=\"b%3C=b%2Acos%28c%2At%29%3C=-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28b%29 ALIGN=MIDDLE  ALT=\"abs%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the amplitude,<BR>\n" );
document.write( "is how far a cabin goes from the middle,<BR>\n" );
document.write( "and that is half the diameter,<BR>\n" );
document.write( "so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28b%29=120m%2F2 ALIGN=MIDDLE  ALT=\"abs%28b%29=120m%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28b%29=60m ALIGN=MIDDLE  ALT=\"abs%28b%29=60m\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28b=-60m%29 ALIGN=MIDDLE  ALT=\"highlight%28b=-60m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE HALF-WAY HEIGHT:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=135m-60m ALIGN=MIDDLE  ALT=\"a=135m-60m\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the height of the wheel hub.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a=75m%29 ALIGN=MIDDLE  ALT=\"highlight%28a=75m%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and you go from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15m ALIGN=MIDDLE  ALT=\"15m\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=135m ALIGN=MIDDLE  ALT=\"135m\"   DISABLED_style= \"max-width:90%; height:auto;\" > and back<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Bb=75m-60m=15m%3C=h%28t%29%3C=75b%2B60m=135m=a-b ALIGN=MIDDLE  ALT=\"a%2Bb=75m-60m=15m%3C=h%28t%29%3C=75b%2B60m=135m=a-b\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE PERIOD:<BR>\n" );
document.write( "Sine and cosine have a period of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "They reach their minimum only once every <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So if you start at the bottom at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "you return there for the first time at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=30 ALIGN=MIDDLE  ALT=\"t=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > minutes,<BR>\n" );
document.write( "at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2A30=2pi ALIGN=MIDDLE  ALT=\"c%2A30=2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=pi%2F15%29 ALIGN=MIDDLE  ALT=\"highlight%28c=pi%2F15%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );