document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=515160>Question 515160</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 17 miles on the second part of the trip at a speed 5 mph slower. the total time for the trip is 2 hrs. What was 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 #343783 by </B><A HREF=/tutors/aboutme.mpl?userid=drcole><B>drcole(72)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=drcole><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=343783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let s represent the speed of the canoeist during the first part of the trip (in miles per hour).  Since the speed is constant over the first part of the trip, we can use the relationship\r<BR>\n" );
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document.write( "Distance = Speed * Time\r<BR>\n" );
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document.write( "to solve for the time it took for the canoeist to travel the first leg in terms of s.  In this case, the distance was 37 miles, so we have the equation\r<BR>\n" );
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document.write( "37 = s * Time<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37%2Fs ALIGN=MIDDLE  ALT=\"37%2Fs\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Time (dividing both sides by s, presuming that s is not 0, which is reasonable since the canoeist traveled 37 miles at that speed)\r<BR>\n" );
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document.write( "So now we have an expression for the amount of time it took to travel the first leg: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+37%2Fs+ ALIGN=MIDDLE  ALT=\"+37%2Fs+\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we move on to the second leg.  According to the question, the canoeist traveled at a speed 5 mph slower than during the first leg of the trip.  We can therefore represent the speed during the second leg as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5 ALIGN=MIDDLE  ALT=\"s+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now we'll solve for the time it took to travel the second leg.  The same relationship between distance, speed, and time applies, but now the distance traveled is 17 miles and the speed is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5 ALIGN=MIDDLE  ALT=\"s+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph.  Solving for time, we get:\r<BR>\n" );
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document.write( "17 = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28s+-+5%29 ALIGN=MIDDLE  ALT=\"%28s+-+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > * Time<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17+%2F+%28s+-+5%29 ALIGN=MIDDLE  ALT=\"17+%2F+%28s+-+5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = Time (dividing both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5 ALIGN=MIDDLE  ALT=\"s+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which we can also assume is non-zero since it is the speed of the canoeist)\r<BR>\n" );
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document.write( "The final bit of information we are given is that the total time of the trip is 2 hours.  We now have the times it took to travel the first and second legs of the trip in term of s, and we can set up an algebraic equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+37%2Fs+%2B+17%2F%28s+-+5%29+=+2+ ALIGN=MIDDLE  ALT=\"+37%2Fs+%2B+17%2F%28s+-+5%29+=+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now we solve for s:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+37+%2B+17s%2F%28s+-+5%29+=+2s+ ALIGN=MIDDLE  ALT=\"+37+%2B+17s%2F%28s+-+5%29+=+2s+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (multiply both sides by s)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+37%28s+-+5%29+%2B+17s+=+2s%28s+-+5%29+ ALIGN=MIDDLE  ALT=\"+37%28s+-+5%29+%2B+17s+=+2s%28s+-+5%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5 ALIGN=MIDDLE  ALT=\"s+-+5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+37s+-+185+%2B+17s+=+2s%5E2+-+10s+ ALIGN=MIDDLE  ALT=\"+37s+-+185+%2B+17s+=+2s%5E2+-+10s+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (distribute the 37 on the left hand side and the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2s ALIGN=MIDDLE  ALT=\"2s\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+54s+-+185+=+2s%5E2+-+10s+ ALIGN=MIDDLE  ALT=\"+54s+-+185+=+2s%5E2+-+10s+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (combine like terms)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+2s%5E2+-+64s+%2B+185+ ALIGN=MIDDLE  ALT=\"+0+=+2s%5E2+-+64s+%2B+185+\"   DISABLED_style= \"max-width:90%; height:auto;\" > (subtract 54s from and add 185 to both sides)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s+=+%2864+%2B-+sqrt%2864%5E2+-+4%2A2%2A185%29%29%2F%282%2A2%29 ALIGN=MIDDLE  ALT=\"+s+=+%2864+%2B-+sqrt%2864%5E2+-+4%2A2%2A185%29%29%2F%282%2A2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (applying the quadratic formula)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s+=+%2864+%2B-+sqrt%282616%29%29%2F4 ALIGN=MIDDLE  ALT=\"+s+=+%2864+%2B-+sqrt%282616%29%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (simplifying)\r<BR>\n" );
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document.write( "So we have two possible solutions for the speed.  Numerically, these two solutions are (to the nearest hundredth):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+%2864+%2B+sqrt%282616%29%29%2F4+=+28.79 ALIGN=MIDDLE  ALT=\"s+=+%2864+%2B+sqrt%282616%29%29%2F4+=+28.79\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+%2864+-+sqrt%282616%29%29%2F4+=+3.21 ALIGN=MIDDLE  ALT=\"s+=+%2864+-+sqrt%282616%29%29%2F4+=+3.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph\r<BR>\n" );
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document.write( "Both are positive, so both might be speeds for the first leg of the trip.  We know, however, that the speed during the second leg of the trip was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+s+-+5+ ALIGN=MIDDLE  ALT=\"+s+-+5+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so:\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+28.79 ALIGN=MIDDLE  ALT=\"s+=+28.79\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5+=+23.79 ALIGN=MIDDLE  ALT=\"s+-+5+=+23.79\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph; and\r<BR>\n" );
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document.write( "if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+=+3.21 ALIGN=MIDDLE  ALT=\"s+=+3.21\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph, then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s+-+5+=+-1.79 ALIGN=MIDDLE  ALT=\"s+-+5+=+-1.79\"   DISABLED_style= \"max-width:90%; height:auto;\" > mph.\r<BR>\n" );
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document.write( "Speed must be a non-negative number, so -1.79 mph cannot be a speed for the second leg of the trip.  Therefore we reject the second solution, giving us that the canoeist traveled at approximately 28.79 mph during the first part of trip, and approximately 23.79 mph during the second part of the trip.  To check, note that the time it took to travel the first leg would therefore be (to the nearest hundredth):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=37%2F28.79+=+1.29 ALIGN=MIDDLE  ALT=\"37%2F28.79+=+1.29\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours\r<BR>\n" );
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document.write( "and the time it took to travel the second leg would be (again, to the nearest hundredth):\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=17%2F23.79+=+0.71 ALIGN=MIDDLE  ALT=\"17%2F23.79+=+0.71\"   DISABLED_style= \"max-width:90%; height:auto;\" > hours\r<BR>\n" );
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document.write( "making the total time traveled 1.29 + 0.71 = 2.00 hours, which is what we wanted. </SPAN>\n" );
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