document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=252614>Question 252614</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can you help me solve this questions, please?\r<BR>\n" );
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document.write( "The original ferris wheel, designed by George Ferris, was 250 feet in diameter. Making 1 revolution for every 40 seconds, h(t)=125sin(0.157t -pi/2) +125 represents the height h (feet) of a seat on the wheel at any time t (seconds). The ride begins when t = 0. During the first 30 seconds of the ride, at what time is a rider on the ferris wheel exactly 125 feet above the ground?\r<BR>\n" );
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document.write( "and also, help me with:\r<BR>\n" );
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document.write( "approximate the solution to 2cosx+x=0 to the nearest hundredth. <BR>\n" );
document.write( "Hints: it should be in radian mode and there are no () around the x+x\r<BR>\n" );
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document.write( "Thanks so much!! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184639 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184639\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=125sin%280.157t+-pi%2F2%29%2B125 ALIGN=MIDDLE  ALT=\"h%28t%29=125sin%280.157t+-pi%2F2%29%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125=125sin%280.157t+-pi%2F2%29%2B125 ALIGN=MIDDLE  ALT=\"125=125sin%280.157t+-pi%2F2%29%2B125\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%28t%29=125 ALIGN=MIDDLE  ALT=\"h%28t%29=125\"   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=125-125=125sin%280.157t+-pi%2F2%29 ALIGN=MIDDLE  ALT=\"125-125=125sin%280.157t+-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 125 from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=125sin%280.157t+-pi%2F2%29 ALIGN=MIDDLE  ALT=\"0=125sin%280.157t+-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%2F125=sin%280.157t+-pi%2F2%29 ALIGN=MIDDLE  ALT=\"0%2F125=sin%280.157t+-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 125.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0=sin%280.157t+-pi%2F2%29 ALIGN=MIDDLE  ALT=\"0=sin%280.157t+-pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%280.157t+-pi%2F2%29=0 ALIGN=MIDDLE  ALT=\"sin%280.157t+-pi%2F2%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t-pi%2F2=arcsin%280%29 ALIGN=MIDDLE  ALT=\"0.157t-pi%2F2=arcsin%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Take the arcsine of both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t-pi%2F2=0%2B2pi%2An ALIGN=MIDDLE  ALT=\"0.157t-pi%2F2=0%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t-pi%2F2=pi%2B2pi%2An ALIGN=MIDDLE  ALT=\"0.157t-pi%2F2=pi%2B2pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > Evaluate the arcsine of 0 to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Don't forget to add in integer multiples 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;\" >\r<BR>\n" );
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document.write( "<font color=red>---------------------------Side Note----------------------</font>\r<BR>\n" );
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document.write( "Take note how if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=0 ALIGN=MIDDLE  ALT=\"n=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we just get 0 or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2pi ALIGN=MIDDLE  ALT=\"2pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3pi ALIGN=MIDDLE  ALT=\"3pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we continue this, we can see that we'll hit all of the integer multiples of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Because of this we can condense the right side to just <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2An ALIGN=MIDDLE  ALT=\"pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" >. \r<BR>\n" );
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document.write( "<font color=red>--------------------------------------------------------</font>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t-pi%2F2=pi%2An ALIGN=MIDDLE  ALT=\"0.157t-pi%2F2=pi%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" > Condense the right side (see side note above).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t=pi%2An%2Bpi%2F2 ALIGN=MIDDLE  ALT=\"0.157t=pi%2An%2Bpi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.157t=%282pi%2An%2Bpi%29%2F2 ALIGN=MIDDLE  ALT=\"0.157t=%282pi%2An%2Bpi%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%28%282pi%2An%2Bpi%29%2F2%29%281%2F0.157%29 ALIGN=MIDDLE  ALT=\"t=%28%282pi%2An%2Bpi%29%2F2%29%281%2F0.157%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F0.157 ALIGN=MIDDLE  ALT=\"1%2F0.157\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate 't'.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%282pi%2An%2Bpi%29%2F0.314 ALIGN=MIDDLE  ALT=\"t=%282pi%2An%2Bpi%29%2F0.314\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply.\r<BR>\n" );
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document.write( "Now let's plug in some values of 'n'. If n=0, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%282pi%2A0%2Bpi%29%2F0.314=pi%2F0.314=10.00507 ALIGN=MIDDLE  ALT=\"t=%282pi%2A0%2Bpi%29%2F0.314=pi%2F0.314=10.00507\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "If n=1, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=%282pi%2A1%2Bpi%29%2F0.314=3pi%2F0.314=30.01522 ALIGN=MIDDLE  ALT=\"t=%282pi%2A1%2Bpi%29%2F0.314=3pi%2F0.314=30.01522\"   DISABLED_style= \"max-width:90%; height:auto;\" >. However, since we're only worried about the first 30 seconds, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=t%3C=30 ALIGN=MIDDLE  ALT=\"0%3C=t%3C=30\"   DISABLED_style= \"max-width:90%; height:auto;\" > which would exclude the 't' value when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=1 ALIGN=MIDDLE  ALT=\"n=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Any other value of 'n' will generate a 't' value outside the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0%3C=t%3C=30 ALIGN=MIDDLE  ALT=\"0%3C=t%3C=30\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=====================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the only solution is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=10.00507 ALIGN=MIDDLE  ALT=\"t=10.00507\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which is approximate) which means that at about 10.00507 seconds, the rider will be 125 feet above the ground during the first 30 seconds.\r<BR>\n" );
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document.write( "===========================================================================\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Since the variable we want to solve for in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Acos%28x%29%2Bx=0 ALIGN=MIDDLE  ALT=\"2%2Acos%28x%29%2Bx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > is buried in a trig function and is outside the trig function, there is no way to solve for it exactly. \r<BR>\n" );
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document.write( "So we have to use a graphing calculator to approximate the roots of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Acos%28x%29%2Bx ALIGN=MIDDLE  ALT=\"2%2Acos%28x%29%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Graphing the given expression, we get\r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2%2Acos%28x%29%2Bx ALIGN=MIDDLE  ALT=\"y=2%2Acos%28x%29%2Bx\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now use the graphing calculator's root/zero function to find the approximate root of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.029866529 ALIGN=MIDDLE  ALT=\"x=-1.029866529\"   DISABLED_style= \"max-width:90%; height:auto;\" > which rounds to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.03 ALIGN=MIDDLE  ALT=\"x=-1.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the nearest hundredth.\r<BR>\n" );
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document.write( "So the answer is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1.03 ALIGN=MIDDLE  ALT=\"x=-1.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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