document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=169618>Question 169618</A>: <I><SPAN STYLE=\"word-break: break-all;\"> During the first part of a trip, a canoeist travels 37 miles at a certain speed. The canoeist travels 7 miles on the second part or the trip at a speed 5 mph slower. The total time for the trip is 4 hours. What is the speed on each part of the trip? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125169 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=125169\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Your basic formula is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=rt ALIGN=MIDDLE  ALT=\"d=rt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, distance equals rate times time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the first leg of the trip, the distance is 37 miles, and both the rate and time are unknown.  Call them <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D ALIGN=MIDDLE  ALT=\"r%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D ALIGN=MIDDLE  ALT=\"t%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 1:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5Dt%5B1%5D=37 ALIGN=MIDDLE  ALT=\"r%5B1%5Dt%5B1%5D=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the second leg of the trip the the distance is 7 miles, and both the rate and time are unknown.  Call them <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B2%5D ALIGN=MIDDLE  ALT=\"t%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 2: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5Dt%5B2%5D=7 ALIGN=MIDDLE  ALT=\"r%5B2%5Dt%5B2%5D=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > which can also be stated Equation 2a: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B2%5D=7%2Fr%5B2%5D ALIGN=MIDDLE  ALT=\"t%5B2%5D=7%2Fr%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We also know that:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 3: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B1%5D=r%5B2%5D%2B5 ALIGN=MIDDLE  ALT=\"r%5B1%5D=r%5B2%5D%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the speed on the second leg was 5 mph slower.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 4: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D%2Bt%5B2%5D=4 ALIGN=MIDDLE  ALT=\"t%5B1%5D%2Bt%5B2%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > because the total time equals 4 hours.  This can also be stated Equation 4a: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t%5B1%5D=4-t%5B2%5D ALIGN=MIDDLE  ALT=\"t%5B1%5D=4-t%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Substituting Equations 3 and 4a into Equation 1 yields:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 5:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5B2%5D%2B5%29%284-t%5B2%5D%29=37 ALIGN=MIDDLE  ALT=\"%28r%5B2%5D%2B5%29%284-t%5B2%5D%29=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "And substituting Equation 2a into equation 5 yields:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Equation 6:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28r%5B2%5D%2B5%29%284-%287%2Fr%5B2%5D%29%29=37 ALIGN=MIDDLE  ALT=\"%28r%5B2%5D%2B5%29%284-%287%2Fr%5B2%5D%29%29=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Expand the binomial:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5B2%5D-7%2B20-%2835%2Fr%5B2%5D%29=37 ALIGN=MIDDLE  ALT=\"4r%5B2%5D-7%2B20-%2835%2Fr%5B2%5D%29=37\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Collect terms:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5B2%5D-%2835%2Fr%5B2%5D%29=24 ALIGN=MIDDLE  ALT=\"4r%5B2%5D-%2835%2Fr%5B2%5D%29=24\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D ALIGN=MIDDLE  ALT=\"r%5B2%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4r%5B2%5D%5E2-24r%5B2%5D-35=0 ALIGN=MIDDLE  ALT=\"4r%5B2%5D%5E2-24r%5B2%5D-35=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the quadratic formula\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D+=+%28-%28-24%29+%2B-+sqrt%28+%28-24%29%5E2-4%2A4%2A%28-35%29+%29%29%2F%282%2A4%29+ ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+%28-%28-24%29+%2B-+sqrt%28+%28-24%29%5E2-4%2A4%2A%28-35%29+%29%29%2F%282%2A4%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D+=+%2824+%2B-+sqrt%28+1136+%29%29%2F8+ ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+%2824+%2B-+sqrt%28+1136+%29%29%2F8+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since 1136 = 16 * 71 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2871%29 ALIGN=MIDDLE  ALT=\"sqrt%2871%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is irrational, reduce to:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%5B2%5D+=+%286+%2B-+sqrt%2871%29%29%2F2 ALIGN=MIDDLE  ALT=\"r%5B2%5D+=+%286+%2B-+sqrt%2871%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2871%29+%3E+6 ALIGN=MIDDLE  ALT=\"sqrt%2871%29+%3E+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286+-+sqrt%2871%29%29%2F2 ALIGN=MIDDLE  ALT=\"%286+-+sqrt%2871%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > result is absurd because, presumably, the canoe never went backwards.  Therefore, the rate of speed for the second leg of the trip is exactly <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286+%2B+sqrt%2871%29%29%2F2 ALIGN=MIDDLE  ALT=\"%286+%2B+sqrt%2871%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and the rate for the first leg of the trip is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286+%2B+sqrt%2871%29%29%2F2%2B5=%286+%2B+sqrt%2871%29%29%2F2%2B10%2F2=%2816+%2B+sqrt%2871%29%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use your calculator to determine <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2871%29 ALIGN=MIDDLE  ALT=\"sqrt%2871%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you want a numerical approximation for the answer, but remember to round off to the nearest whole number because all of your given values, i.e. the two distances and the total time were given as whole numbers.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Calculating the times for the indiviual legs and using these values to check the answer is left as an exercise for the student.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );