document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=322397>Question 322397</A>: <I><SPAN STYLE=\"word-break: break-all;\"> To deliver a package, a messenger must travel at a speed of 60 miles per hour on land and then use a motorboat whose speed is 20 miles per hour in still water. The messenger goes by land to a dock and then travels on a river against a current of 4 miles per hour. He reaches his destination in 4.5 hours and then returns to the starting point in 3.5 hours. How far did the messenger travel by land and how far by water? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #230795 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=230795\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "Everything in this problem is based on the <i>distance equals rate times time</i> formula.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_l\"> represent the distance traveled on land.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_w\"> represent the distance traveled on the water.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> represent the elapsed time traveling on land.\r<BR>\n" );
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document.write( "We know the rate of speed on land because that is given as 60 mph.\r<BR>\n" );
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document.write( "The rate of speed for the outbound trip on water is the rate in still water MINUS the rate of the current (because he was traveling AGAINST the current), so 20 mph minus 4 mph is 16 mph.\r<BR>\n" );
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document.write( "Likewise, the rate of speed for the return trip on water is 24 mph.\r<BR>\n" );
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document.write( "If the total outbound trip took 4.5 hours and the land part of the trip took <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> hours, then the water part of the outbound trip must have taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4.5\ -\ t\"> hours.\r<BR>\n" );
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document.write( "Similarly, the water part of the return trip must have taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 3.5\ -\ t\"> hours.\r<BR>\n" );
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document.write( "Let's put some of this together.  We can describe the outbound trip over water by the relationship:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 16(4.5\ -\ t)\">\r<BR>\n" );
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document.write( "And we can describe the return trip over water as:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 24(3.5\ -\ t)\">\r<BR>\n" );
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document.write( "Having two things both equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_w\">, we can set them equal to each other:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 16(4.5\ -\ t)\ =\ 24(3.5\ -\ t)\">\r<BR>\n" );
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document.write( "From which we can solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\ =\ 1.5\"> hours, the travel time on land.  (Verification of this result is left as an exercise for the student).\r<BR>\n" );
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document.write( "Then, knowing that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_l\ =\ 60t\">, we can say that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_l\ = 90\">\r<BR>\n" );
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document.write( "Further, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\ =\ 1.5\"> says that the travel time for the outbound trip on water is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4.5\ -\ 1.5\ =\ 3\">, hence the distance over water for the outbound trip is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d_w\ =\ 16\,\cdot\,3\ =\ 48\"> miles.\r<BR>\n" );
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document.write( "As a check, the distance over water for the return trip better be the same as for the outbound trip:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ d_w\ =\ 24(3.5\ -\ 1.5)\ =\ 48\">\r<BR>\n" );
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document.write( "And our answer checks.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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