document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=682777>Question 682777</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help on this calculus word problem:\r<BR>\n" );
document.write( "\n" );
document.write( "Two ants are at a common point at time t=0, the first ant starts crawling along a straight line at the rate of 4 ft/min. Two minutes later, the second ant starts crawling in a direction perpendicular to that of the first, at a rate of 5 ft/min. How fast is the distance between them changing when the first insect has traveled 12 feet?\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 #423333 by </B><A HREF=/tutors/aboutme.mpl?userid=funmath><B>funmath(2933)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=funmath><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=423333\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Two ants are at a common point at time t=0, the first ant starts crawling along a straight line at the rate of 4 ft/min. Two minutes later, the second ant starts crawling in a direction perpendicular to that of the first, at a rate of 5 ft/min. How fast is the distance between them changing when the first insect has traveled 12 feet?\r<BR>\n" );
document.write( "\n" );
document.write( "<br> Let r be the distance between them, x be the distance of the first and y be the distance of the second. Then according to the Pythagorean theorem:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"r%5E2=x%5E2%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Taking the derivative of the equation with respect to time yields:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2r%28dr%2Fdt%29=2x%28dx%2Fdt%29%2B2y%28dy%2Fdt%29 ALIGN=MIDDLE  ALT=\"2r%28dr%2Fdt%29=2x%28dx%2Fdt%29%2B2y%28dy%2Fdt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Everything is divisible by 2 so divide the whole thing by 2 to make things easier.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28r%28dr%2Fdt%29=x%28dx%2Fdt%29%2By%28dy%2Fdt%29%29 ALIGN=MIDDLE  ALT=\"highlight%28r%28dr%2Fdt%29=x%28dx%2Fdt%29%2By%28dy%2Fdt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now use a little algebra to find the missing components.<BR>\n" );
document.write( "Given:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt=4 ALIGN=MIDDLE  ALT=\"dx%2Fdt=4\"   DISABLED_style= \"max-width:90%; height:auto;\" >the rate of the first ant<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdt=5 ALIGN=MIDDLE  ALT=\"dy%2Fdt=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >the rate of the second ant<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=12 ALIGN=MIDDLE  ALT=\"x=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > the distance the first ant traveled.<BR>\n" );
document.write( "<br>distance = rate*time<br> \r<BR>\n" );
document.write( "\n" );
document.write( "We get t from the first ant.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12=4t ALIGN=MIDDLE  ALT=\"12=4t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F4=4t%2F4 ALIGN=MIDDLE  ALT=\"12%2F4=4t%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3=t ALIGN=MIDDLE  ALT=\"3=t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Therefore the second ant traveled.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%28t-2%29 ALIGN=MIDDLE  ALT=\"y=5%28t-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5%28highlight%283%29-2%29 ALIGN=MIDDLE  ALT=\"y=5%28highlight%283%29-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=5 ALIGN=MIDDLE  ALT=\"y=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "<br> We use the Pythagorean Theorem to find r <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=x%5E2%2By%5E2 ALIGN=MIDDLE  ALT=\"r%5E2=x%5E2%2By%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=12%5E2%2B5%5E2 ALIGN=MIDDLE  ALT=\"r%5E2=12%5E2%2B5%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=144%2B25 ALIGN=MIDDLE  ALT=\"r%5E2=144%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5E2=169 ALIGN=MIDDLE  ALT=\"r%5E2=169\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=sqrt%28169%29 ALIGN=MIDDLE  ALT=\"r=sqrt%28169%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r=13 ALIGN=MIDDLE  ALT=\"r=13\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "Take all the parts and put them into the equation you took the derivative of.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28r%28dr%2Fdt%29=x%28dx%2Fdt%29%2By%28dy%2Fdt%29%29 ALIGN=MIDDLE  ALT=\"highlight%28r%28dr%2Fdt%29=x%28dx%2Fdt%29%2By%28dy%2Fdt%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%28dr%2Fdt%29=12%284%29%2B5%285%29 ALIGN=MIDDLE  ALT=\"13%28dr%2Fdt%29=12%284%29%2B5%285%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%28dr%2Fdt%29=48%2B25 ALIGN=MIDDLE  ALT=\"13%28dr%2Fdt%29=48%2B25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%28dr%2Fdt%29=73 ALIGN=MIDDLE  ALT=\"13%28dr%2Fdt%29=73\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt=73%2F13 ALIGN=MIDDLE  ALT=\"dr%2Fdt=73%2F13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dr%2Fdt=5.6 ALIGN=MIDDLE  ALT=\"dr%2Fdt=5.6\"   DISABLED_style= \"max-width:90%; height:auto;\" >ft/min\r<BR>\n" );
document.write( "\n" );
document.write( "That should be it barring any careless mistakes. Happy Calculating! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );