document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=180490>Question 180490</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 4.	The Hudson River flows at a rate of 5 miles per hour.  A patrol boat travels 40 miles upriver, and returns in a total time of 6 hours.  What is the speed of the boat 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 #135261 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=135261\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "This is a distance, rate, and time problem, so the basic formula to use is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large d = rt\">.  However, we will find it convenient to express this relationship as <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t = \frac {d}{r}\"> for this particular problem.\r<BR>\n" );
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document.write( "We are asked to find the rate of the boat in still water, so let's call that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r_s\"> to distinguish it from the <i>r</i> in the basic formula above.  We are given the speed of the current, 5 mph, so we know that the speed of the boat relative to the shoreline as the boat goes upstream is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r_s - 5\"> and the speed of the boat relative to the shoreline as the boat goes downstream is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r_s + 5\">.  We also know that each leg of the trip is 40 miles long.  Lastly, let's say that the time it takes for the boat to make the upstream trip is <i>t</i> hours.  That means that, since the entire trip took 6 hours, the downstream trip must have taken 6 - <i>t</i> hours.\r<BR>\n" );
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document.write( "Now let's use the formula, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large t = \frac {d}{r}\"> to describe the upstream trip:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math t = \frac {40}{r_s - 5} \">\r<BR>\n" );
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document.write( "And the downstream trip:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math 6 - t = \frac {40}{r_s + 5}\">\r<BR>\n" );
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document.write( "First, solve the downstream equation for <i>t</i>:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math t = 6 - \left(\frac {40}{r_s + 5}\right)\">\r<BR>\n" );
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document.write( "And simplify the result:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math t = \frac {\[6(r_s + 5)\] - 40}{r_s + 5}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math t = \frac {6r_s - 10}{r_s + 5}\">\r<BR>\n" );
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document.write( "Next if you compare the simplified form of the downstream equation with the upstream equation, you will notice that we have two different expressions in <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large r_s\"> that are both equal to the same thing, namely <i>t</i>.  We can now set these two expressions equal to each other:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math \frac {40}{r_s - 5} = \frac {6r_s - 10}{r_s + 5} \">\r<BR>\n" );
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document.write( "Put everything on the left and apply the common denominator <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {r_s}^2 - 25\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math \frac {\[40(r_s + 5)\] - \[(6r_s - 10)(r_s - 5)\]}{{r_s}^2 - 25} = 0 \">\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large {r_s}^2 - 25\">:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math \[40(r_s + 5)\] - \[(6r_s - 10)(r_s - 5)\] = 0 \">\r<BR>\n" );
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document.write( "Apply the distributive property and collect terms:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math 40r_s + 200 - (6{r_s}^2 - 30r_s - 10r_s + 50) = 0 \">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math -6{r_s}^2 + 80r_s + 150 = 0 \">\r<BR>\n" );
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document.write( "Multiply by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large -\frac {1}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math 3{r_s}^2 - 40r_s - 75 = 0 \">\r<BR>\n" );
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document.write( "Factor:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math (3r_s + 5)(r_s - 15) = 0 \">\r<BR>\n" );
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document.write( "Therefore <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math r_s = -\frac {5}{3} \"> or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math r_s = 15 \">\r<BR>\n" );
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document.write( "The negative answer is absurd because we know the boat wasn't going backwards, so exclude that answer as an extraneous root introduced by the action of squaring the variable during the solution process, and that leaves us with  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {          }\math r_s = 15 \">\r<BR>\n" );
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document.write( "Check the answer.\r<BR>\n" );
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document.write( "If the speed in still water is 15 mph, then the upstream speed must have been 15 - 5 = 10 mph.  At 10 mph, it takes <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 40 \div 10 = 4\"> hours to make the trip.  The downstream speed must have been 15 + 5 = 20 mph.  At 20 mph, it takes <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 40 \div 20 = 2\"> hours to make the trip.  4 hours plus 2 hours = 6 hours which was the given elapsed time.  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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