document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=759069>Question 759069</A>: <I><SPAN STYLE=\"word-break: break-all;\"> two straight paths,inclined to one another at 60 degrees,interesct at a point o.A boy A is on one path 300m from O while a boy B is on the other path 400m from O.angle AOB=60 degrees.Simultaneously the boys begin to run towards O,A with speed 15 kmh-1 and B with speed 12 kmh-1 What is the shortest distance between the two boys?\r<BR>\n" );
document.write( "\n" );
document.write( "I calculated Va-Vb=360.6<BR>\n" );
document.write( "I couldn't calculate the shortest distance (l) as I do not know the angle between Ab and Va-Vb, </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #461974 by </B><A HREF=/tutors/aboutme.mpl?userid=Alan3354><B>Alan3354(69443)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Alan3354><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=Alan3354><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=461974\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> two straight paths,inclined to one another at 60 degrees,interesct at a point o.A boy A is on one path 300m from O while a boy B is on the other path 400m from O.angle AOB=60 degrees.Simultaneously the boys begin to run towards O,A with speed 15 kmh-1 and B with speed 12 kmh-1 What is the shortest distance between the two boys?<BR>\n" );
document.write( "----------------------------<BR>\n" );
document.write( "The speed of A = 250 meters/min<BR>\n" );
document.write( "Speed of B = 200 m/min<BR>\n" );
document.write( "--------<BR>\n" );
document.write( "a & b = distance from O at time t (in minutes)<BR>\n" );
document.write( "a = 300 - 250t<BR>\n" );
document.write( "b = 400 - 200t<BR>\n" );
document.write( "-------<BR>\n" );
document.write( "d = distance between A & B at time t<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+a%5E2+%2B+b%5E2+-+2ab%2Acos%2860%29 ALIGN=MIDDLE  ALT=\"d%5E2+=+a%5E2+%2B+b%5E2+-+2ab%2Acos%2860%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+a%5E2+%2B+b%5E2+-+ab ALIGN=MIDDLE  ALT=\"d%5E2+=+a%5E2+%2B+b%5E2+-+ab\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "------<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+2500%2A%2821t%5E2+-+60t+%2B+52%29 ALIGN=MIDDLE  ALT=\"d%5E2+=+2500%2A%2821t%5E2+-+60t+%2B+52%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Without calculus:<BR>\n" );
document.write( "Finding the minimum for d is the same as finding the minimum for d^2<BR>\n" );
document.write( "The min is the vertex of the parabola <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5E2+=+2500%2A%2821t%5E2+-+60t+%2B+52%29 ALIGN=MIDDLE  ALT=\"d%5E2+=+2500%2A%2821t%5E2+-+60t+%2B+52%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The vertex is on the LOS, Line of Symmetry t = -b/2a<BR>\n" );
document.write( "t = 60/42 = 10/7 seconds<BR>\n" );
document.write( "Sub for t as below<BR>\n" );
document.write( "==================================<BR>\n" );
document.write( "Using the 1st derivative:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%28t%29+=+50%2A%2821t%5E2+-+60t+%2B+52%29%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"d%28t%29+=+50%2A%2821t%5E2+-+60t+%2B+52%29%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "------<BR>\n" );
document.write( "d'(t) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2A%281%2F2%29%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%2842t+-+60%29 ALIGN=MIDDLE  ALT=\"50%2A%281%2F2%29%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%2842t+-+60%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "d'(t) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=150%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%287t+-+10%29 ALIGN=MIDDLE  ALT=\"150%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%287t+-+10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "@ d'(t) = 0:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=150%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%287t+-+10%29+=+0 ALIGN=MIDDLE  ALT=\"150%2A%2821t%5E2+-+60t+%2B+52%29%5E%28-1%2F2%29%2A%287t+-+10%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7t+-+10+=+0 ALIGN=MIDDLE  ALT=\"7t+-+10+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "t = 10/7 minutes<BR>\n" );
document.write( "-----------------<BR>\n" );
document.write( "d(10/7) = 151.186 meters<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );