document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151632>Question 1151632</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A boat’s speed in still in water is 4km/h.The boat cruises from A to B along a river flowing at an average speed of X km/h in the direction A to B . If the distance AB is 5 km and the boat takes 2hr more on its return journey,determine x. Hence,find the total time taken for the whole journey. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #773458 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=773458\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let 4 = the rate of the boat in still water.<BR>\n" );
document.write( "let R = the rate of the river.<BR>\n" );
document.write( "let distance = 5<BR>\n" );
document.write( "rate * time = distance.<BR>\n" );
document.write( "with the current, the rate becomes (4 + R)<BR>\n" );
document.write( "against the current, the rate becomes (4 - R)<BR>\n" );
document.write( "let T = the time it takes going with the current.<BR>\n" );
document.write( "then T + 2 = the time it takes coming back against the current.\r<BR>\n" );
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document.write( "the formulas are:<BR>\n" );
document.write( "(4 + R) * T = 5<BR>\n" );
document.write( "(4 - R) * (T + 2) = 5\r<BR>\n" );
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document.write( "simplify these equations to get:<BR>\n" );
document.write( "4T + RT = 5<BR>\n" );
document.write( "4T + 8 - RT - 2R = 5\r<BR>\n" );
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document.write( "since they both equal 5, set them equal to each other to get:<BR>\n" );
document.write( "4T + RT = 4T + 8 - RT - 2R<BR>\n" );
document.write( "the 4T cancels out and you get RT = 8 - RT - 2R<BR>\n" );
document.write( "subtract the right side of the equation from both sides of the equation to get:<BR>\n" );
document.write( "RT - 8 + RT + 2R = 0<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "2RT + 2R - 8 = 0\r<BR>\n" );
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document.write( "factor the first original equation of 4T + RT = 5 to get:<BR>\n" );
document.write( "T * (R + 4) = 5<BR>\n" );
document.write( "divide both sides of this equation by (R + 4) to get:<BR>\n" );
document.write( "T = 5 / (R + 4)\r<BR>\n" );
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document.write( "replace T in 2RT + 2R - 8 = 0 to get:<BR>\n" );
document.write( "2R * 5 / (R + 4) + 2R - 8 = 0<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "10R / (R + 4) + 2R - 8 = 0<BR>\n" );
document.write( "multiply both sides of this equation by (R + 4) to get:<BR>\n" );
document.write( "10R + 2R * (R + 4) - 8 * (R + 4) = 0<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "10R + 2R^2 + 8R - 8R - 32 = 0<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "2R^2 + 10R - 32 = 0<BR>\n" );
document.write( "divide both sides of this equation by 2 to get:<BR>\n" );
document.write( "R^2 + 5R - 16 = 0\r<BR>\n" );
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document.write( "factor this quadratic to get:<BR>\n" );
document.write( "R = 2.2169905660283 or R = -7.2169905660283<BR>\n" );
document.write( "R has to be positive, so R = 2.2169905660283\r<BR>\n" );
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document.write( "your original equations are:<BR>\n" );
document.write( "(4 + R) * T = 5<BR>\n" );
document.write( "(4 - R) * (T + 2) = 5\r<BR>\n" );
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document.write( "when R = 2.2169905660283, the first equation becomes:<BR>\n" );
document.write( "(4 + 2.2169905660283) * T = 5<BR>\n" );
document.write( "solve for T to get:<BR>\n" );
document.write( "T = 5 / 6.2169905660283 = .8042476415\r<BR>\n" );
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document.write( "your two original equations are, once again:<BR>\n" );
document.write( "(4 + R) * T = 5<BR>\n" );
document.write( "(4 - R) * (T + 2) = 5\r<BR>\n" );
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document.write( "when R = 2.2169905660283 and T = .8042476415, these equations become:<BR>\n" );
document.write( "(4 + 2.2169905660283) * .8042476415 = 5<BR>\n" );
document.write( "(4 - 2.2169905660283) * (.8042476415 + 2) = 5\r<BR>\n" );
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document.write( "use your calculator to confirm these equations are true.<BR>\n" );
document.write( "i confirmed using my calculator.\r<BR>\n" );
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document.write( "your solution is that the total time taken for the whole journey is .8042476415 + .8042476415 + 2 = 3.608495283 hours.\r<BR>\n" );
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document.write( "it's not exactly a clean answer, but it does check out, assuming my original equations are correct.<BR>\n" );
document.write( "i believe they are:\r<BR>\n" );
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document.write( "i used the following quadratic equation solver to find the roots of the quadratic.<BR>\n" );
document.write( "https://www.mathsisfun.com/quadratic-equation-solver.html<BR>\n" );
document.write( "this would have been a huge chore otherwise.<BR>\n" );
document.write( "i didn't want to go through the large manual effort only to find that the solution was not good.<BR>\n" );
document.write( "it turned out that the answer was good, even if it was messy.\r<BR>\n" );
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document.write( "the basic concepts is:<BR>\n" );
document.write( "when you're going with the current, the formula becomes (B + R) * T = 5<BR>\n" );
document.write( "when you're going against the current, the formula becomes (B - R) * (T + 2) = 5 </SPAN>\n" );
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