document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=919886>Question 919886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Towns A and B are 540km apart. A train leaves A and travels to B at a constant speed. A second train leaves A 1h later and, after 2h, reaches a point passed by the first train 40min earlier. The second train then increases speed by 5km/h and arrives at B at the same time as the first train. What were the speeds of the trains when they left A?<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 #558278 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=558278\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  Towns A and B are 540km apart.<BR>\n" );
document.write( " A train leaves A and travels to B at a constant speed.<BR>\n" );
document.write( " A second train leaves A 1h later and, after 2h, reaches a point passed by the first train 40min earlier.<BR>\n" );
document.write( " The second train then increases speed by 5km/h and arrives at B at the same time as the first train.<BR>\n" );
document.write( " What were the speeds of the trains when they left A?<BR>\n" );
document.write( ":<BR>\n" );
document.write( "A-----------------p-----------540 km-----------------------------B<BR>\n" );
document.write( ":<BR>\n" );
document.write( "1st train got to p in 3 hr - 40 min or 2<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > hrs<BR>\n" );
document.write( "2nd train got to p in 2 hrs, (after leaving 1 hr later than the 1st train)<BR>\n" );
document.write( "We can say that 2nd train travels the same dist in 2 hrs, as the 1st train in 2<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > hrs,<BR>\n" );
document.write( " time ratio; 1st train to 2nd train 2.33:2 or in integers 7:6 ratio<BR>\n" );
document.write( ":<BR>\n" );
document.write( "let s = speed of the 1st train<BR>\n" );
document.write( "then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=540%2Fs ALIGN=MIDDLE  ALT=\"540%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" > = travel time of the 1st train<BR>\n" );
document.write( ":<BR>\n" );
document.write( "speed varies inversely with time therefore<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F6 ALIGN=MIDDLE  ALT=\"7%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >s = initial speed of 2nd train<BR>\n" );
document.write( "and<BR>\n" );
document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F6 ALIGN=MIDDLE  ALT=\"7%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >s + 5) = 2nd train speed from p to B<BR>\n" );
document.write( "The distance to p (dist = speed * time) = 2*<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F6 ALIGN=MIDDLE  ALT=\"7%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >s = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F3 ALIGN=MIDDLE  ALT=\"7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >s<BR>\n" );
document.write( ":<BR>\n" );
document.write( "2nd trains time to p = 3 hrs after the 1st train started<BR>\n" );
document.write( "Write a time equation<BR>\n" );
document.write( "3 hrs to p + time traveling from p to B = 1st train travel time<BR>\n" );
document.write( "3 + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28540-%287%2F3%29s%29%29%2F%28%28%287%2F6%29s%2B5%29%29 ALIGN=MIDDLE  ALT=\"%28%28540-%287%2F3%29s%29%29%2F%28%28%287%2F6%29s%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=540%2Fs ALIGN=MIDDLE  ALT=\"540%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It's such a pain in the you know what, to deal with these fractions, try to use decimals and see what we get<BR>\n" );
document.write( "3 + <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28540-2.333s%29%29%2F%28%281.167s%2B5%29%29 ALIGN=MIDDLE  ALT=\"%28%28540-2.333s%29%29%2F%28%281.167s%2B5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=540%2Fs ALIGN=MIDDLE  ALT=\"540%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "multiply by s(1.167s+5)<BR>\n" );
document.write( "3s(1.167s+5) + s(540-2.33s) = 540(1.167s+5)<BR>\n" );
document.write( "3.5s^2 + 15s + 540s - 2.33s^2 = 630s + 2700<BR>\n" );
document.write( "Combine like terms<BR>\n" );
document.write( "3.5s^2 - 2.33s^2 - 630s + 15s + 540s - 2700 = 0<BR>\n" );
document.write( "1.167s^2 - 75s - 2700 = 0<BR>\n" );
document.write( "using the quadratic formula, I got a postive solution of <BR>\n" );
document.write( "s = 90 km/hr is the speed of the 1st train<BR>\n" );
document.write( "then<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2F6 ALIGN=MIDDLE  ALT=\"7%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >*90 = 105 km/hr, initial speed of the 2nd train<BR>\n" );
document.write( ":<BR>\n" );
document.write( ":<BR>\n" );
document.write( "Let's see if we lucked out and got this right, find the times.<BR>\n" );
document.write( "1st train travel time = 540/90 = 6 hrs<BR>\n" );
document.write( "2nd train travel time to p<BR>\n" );
document.write( "2 * 105 = 210 km to p<BR>\n" );
document.write( "2nd train speed to B = 110 km/hr (5 km/hr more)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28540-210%29%2F110 ALIGN=MIDDLE  ALT=\"%28540-210%29%2F110\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 3 hrs<BR>\n" );
document.write( "2nd train total time<BR>\n" );
document.write( "1 hr before starting<BR>\n" );
document.write( "2 hrs to p<BR>\n" );
document.write( "3 hrs p to B<BR>\n" );
document.write( "------------<BR>\n" );
document.write( "6 hrs, same as 1st train!!!\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );