document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=994014>Question 994014</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A particle is moving around the ellipse 4x^2+16y^2 = 64. At any time t its x and y coordinates are given by x = 4cos(t) and y = 2sin(t). At what rate is the particle's distance to the origin changing when t = &#960;/4?\r<BR>\n" );
document.write( "\n" );
document.write( "I know this is a related rates problem. But I don't know how to set this problem up in order to solve. \r<BR>\n" );
document.write( "\n" );
document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #613214 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=613214\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I would think this is a calculus problem.<BR>\n" );
document.write( "The distance to the origin as a function of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > (time) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%28t%29=sqrt%28x%5E2%2By%5E2%29=sqrt%28%284cos%28t%29%29%5E2%2B%282sin%28t%29%29%5E2%29=sqrt%2816%28cos%28t%29%29%5E2%2B4%28sin%28t%29%29%5E2%29=sqrt%284%284%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2%29%29=2sqrt%284%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The function showing the rate of change at 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;\" > is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dD%2Fdt ALIGN=MIDDLE  ALT=\"dD%2Fdt\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and it can be calculated using the chain rule<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%28t%29=2%2AF%28G%28t%29%29 ALIGN=MIDDLE  ALT=\"D%28t%29=2%2AF%28G%28t%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%28u%29=sqrt%28u%29=u%5E%28%221%2F2%22%29 ALIGN=MIDDLE  ALT=\"F%28u%29=sqrt%28u%29=u%5E%28%221%2F2%22%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u=G%28t%29=4%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2 ALIGN=MIDDLE  ALT=\"u=G%28t%29=4%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dF%2Fdu=%281%2F2%29u%5E%28%22-1%2F2%22%29=1%2F%282u%5E%28%221%2F2%22%29%29=1%2F2sqrt%28u%29 ALIGN=MIDDLE  ALT=\"dF%2Fdu=%281%2F2%29u%5E%28%22-1%2F2%22%29=1%2F%282u%5E%28%221%2F2%22%29%29=1%2F2sqrt%28u%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28sqrt%284%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2%29%29%2Fdt=d%284%28cos%28t%29%29%5E2%29%2Fdt%2Bd%28%28sin%28t%29%29%5E2%29%2Fdt ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Applying the chain rule again, and again, to both terms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%284%28cos%28t%29%29%5E2%29%2Fdt=4%2A%282cos%28t%29%29d%28cos%28t%29%29%2Fdt=4%2A2%2Acos%28t%29%28-sin%28t%29%29=-8cos%28t%29sin%28t%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28%28sin%28t%29%29%5E2%29%2Fdt=%282sin%28t%29%29d%28sin%28t%29%29%2Fdt=2%2Asin%28t%29%28cos%28t%29%29=2cos%28t%29sin%28t%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28sqrt%284%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2%29%29%2Fdt=-8cos%28t%29sin%28t%29%2B2cos%28t%29sin%28t%29=-6cos%28t%29sin%28t%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dD%2Fdt=2%281%2F%282sqrt%28u%29%29%29%28-6cos%28t%29sin%28t%29%29=%28-6cos%28t%29sin%28t%29%29%2Fsqrt%28u%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Substituting the expression for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u ALIGN=MIDDLE  ALT=\"u\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28t%29=dD%2Fdt=%28-6cos%28t%29sin%28t%29%29%2Fsqrt%284%28cos%28t%29%29%5E2%2B%28sin%28t%29%29%5E2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "For <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=pi%2F4 ALIGN=MIDDLE  ALT=\"t=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=pi%2F4 ALIGN=MIDDLE  ALT=\"t=pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28cos%28t%29=cos%28pi%2F4%29=sqrt%282%29%2F2%2Ccos%28t%29=cos%28pi%2F4%29=sqrt%282%29%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=%28-6cos%28pi%2F4%29sin%28pi%2F4%29%29%2Fsqrt%284%28cos%28pi%2F4%29%29%5E2%2B%28sin%28pi%2F4%29%29%5E2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=%28-6%28sqrt%282%29%2F2%29%28sqrt%282%29%2F2%29%2Fsqrt%284%28sqrt%282%29%2F2%29%5E2%2B%28sqrt%282%29%2F2%29%5E2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=%28-6%282%2F4%29%29%2Fsqrt%284%282%2F4%29%2B%282%2F4%29%29%29%29 ALIGN=MIDDLE  ALT=\"rate%28pi%2F4%29=%28-6%282%2F4%29%29%2Fsqrt%284%282%2F4%29%2B%282%2F4%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=-3%2Fsqrt%285%2F2%29 ALIGN=MIDDLE  ALT=\"rate%28pi%2F4%29=-3%2Fsqrt%285%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=-3sqrt%282%29%2Fsqrt%285%29 ALIGN=MIDDLE  ALT=\"rate%28pi%2F4%29=-3sqrt%282%29%2Fsqrt%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rate%28pi%2F4%29=-3sqrt%2810%29%2F5 ALIGN=MIDDLE  ALT=\"rate%28pi%2F4%29=-3sqrt%2810%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );