document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199336>Question 199336</A>: <I><SPAN STYLE=\"word-break: break-all;\"> DISTANCE, SPEED, AND TIME: It took a crew two hours and 40mins. to row 6km upstream and back again. If the rate of flow of the stream was 3km/h, what was the rowing speed of the crew in still water? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149797 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=149797\">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( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> represent the rate in still 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 for the upstream part of the trip.\r<BR>\n" );
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document.write( "When the boat is going upstream, that is, against the 3 km/hr current, the speed relative to the bank of the river must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r - 3\">.  Conversely, the speed relative to the bank of the river must be <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r + 3\"> on the downstream part of the trip.\r<BR>\n" );
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document.write( "The total elapsed time for both legs of the trip was 2 hours and 40 minutes.  Since 40 minutes is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{2}{3}\"> hour, the total trip was <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2\frac{2}{3} = \frac{8}{3}\"> hour.  That means that if the upstream trip took <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\"> hours, then the downstream trip must have taken <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{8}{3} - t\"> hours.\r<BR>\n" );
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document.write( "Since we know that distance equals rate times time, or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d = rt\">, which can be expressed as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t = \frac{d}{r}\">, and we know that the distance for both the upstream and downstream trips is 6 km, we can describe the upstream trip by the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{6}{r-3}\">\r<BR>\n" );
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document.write( "Similarly, the downstream trip can be described:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \frac{8}{3}-t = \frac{6}{r+3}\">\r<BR>\n" );
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document.write( "which can be arranged so that it is expressed in terms of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ t = \frac{6}{r + 3} - \frac{8}{3} = \frac{8r+6}{3(r + 3)}\">\r<BR>\n" );
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document.write( "Now that we have two different expressions in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r\"> that are equal to <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t\">, 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 \ \ \ \ \ \ \ \ \ \ \frac{6}{r-3} =  \frac{8r+6}{3(r + 3)}\">\r<BR>\n" );
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document.write( "Solve the proportion by first cross-multiplying:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 8r^2 - 18r - 18 = 18r + 54\">\r<BR>\n" );
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document.write( "Collect terms, put the quadratic into standard form, and eliminate the factor common to the coefficients, i.e. 4:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2r^2 - 9t - 18 = 0\">\r<BR>\n" );
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document.write( "Now all you have to do is solve the quadratic by ordinary means (this one factors rather tidily).  You will find two roots.  Exclude the negative root because negative rate makes little sense in this regard.  The positive root will be the answer to the question posed.\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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