document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=208387>Question 208387</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. The speed of a boat in still water is 10 mph. If it travels on a river 6 miles downstream in the same amount of time it takes to travel 3 miles upstream, what is the speed of the current?<BR>\n" );
document.write( "2. The speed of a boat in still water is 8 mph. If it travels 24 miles upstream and 24 miles downstream in a total of 12 hours, what is the speed of the current?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157725 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=157725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> PROBLEM NUMBER 1<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "let b = speed of the boat in still water.<BR>\n" );
document.write( "let c = speed of the current<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "rate * time = distance<BR>\n" );
document.write( "let h = time<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "going downstream, the formula is (b + c) * h = 6<BR>\n" );
document.write( "going upstream, the formula is (b - c) * h = 3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since b = 10, these formula become:<BR>\n" );
document.write( "(10 + c) * h = 6<BR>\n" );
document.write( "and<BR>\n" );
document.write( "(10 - c) * h = 3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since 6 = 2 * 3, then we can substitute (10 - c) * h for 3 to get:<BR>\n" );
document.write( "(10 + c) * h = (10 - c) * h * 2<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this formula becomes:<BR>\n" );
document.write( "10h + ch = 20h - 2ch<BR>\n" );
document.write( "we add 2ch to both sides and we subtract 10h from both sides to get:<BR>\n" );
document.write( "3ch = 10h<BR>\n" );
document.write( "we divide both sides by h to get:<BR>\n" );
document.write( "3c = 10<BR>\n" );
document.write( "we divide both sides by 3 to get:<BR>\n" );
document.write( "c = 10/3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "our first equation is:<BR>\n" );
document.write( "(10 + c) * h = 6<BR>\n" );
document.write( "we replace c with 10/3 to get:<BR>\n" );
document.write( "(10 + 10/3) * h = 6<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "(40/3) * h = 6<BR>\n" );
document.write( "we multiply both sides by 3 to get:<BR>\n" );
document.write( "40 * h = 18<BR>\n" );
document.write( "we divide both sides by 40 to get:<BR>\n" );
document.write( "h = 18/40<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "that's our answer.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to prove the answer is correct, we substitute in the original formulas:<BR>\n" );
document.write( "(10 + 10/3) * 18/40 = 6<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "(40/3) * 18/40 = 6<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "18/3 = 6 which becomes 6 = 6 which is true.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "(10 - 10/3) * 18/40 = 3<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "(20/3) * 18/40 = 3<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "(1/3) * (18/2) = 3<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "(1/3) * 9 = 3 which becomes 3 = 3 which is true3.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "answer checks out so h = 18/40 is good.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "PROBLEM NUMBER 2<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "b = speed of boat = 8 miles per hour<BR>\n" );
document.write( "c = speed of current<BR>\n" );
document.write( "f = time to go with the current<BR>\n" );
document.write( "g = time to go against the current<BR>\n" );
document.write( "d = distance = 24<BR>\n" );
document.write( "time to go down and back = 12 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "(b-c)*g = 24<BR>\n" );
document.write( "(b+c)*f = 24<BR>\n" );
document.write( "g + f = 12<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "(b+c)*f = (b-c)*g because they both equal to 24.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this expands to:<BR>\n" );
document.write( "bf + cf = bg - cg<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since f+g = 12, then f = 12-g<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "substitute 12-g for f in bf + cf = bg - cg to get:<BR>\n" );
document.write( "b * (12 - g) + c * (12 - g) = b*g - c*g<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since b = 8, substitute 8 for b in equation to get:<BR>\n" );
document.write( "8 * (12 - g) + c * (12 - g) = 8*g - c*g<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "expand to get:<BR>\n" );
document.write( "96 - 8g + 12c - cg = 8g - cg<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "subtract 8g from both sides and add cg to both sides to get:<BR>\n" );
document.write( "96 - 16g + 12c = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "solving for c, we get:<BR>\n" );
document.write( "c = (16g - 96)/12<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "take the original equation of (b-c) * g = 24<BR>\n" );
document.write( "substitute 8 for b<BR>\n" );
document.write( "substitute (16g-96)/12 for c<BR>\n" );
document.write( "solve for g<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "equation becomes:<BR>\n" );
document.write( "(8 - ((16g-96)/12))*g = 24<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "multiply both sides by 12 to get:<BR>\n" );
document.write( "(96 - (16g - 96))*g = 24 * 12<BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "96g - 16g^2 + 96g = 288<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "combine like terms to get:<BR>\n" );
document.write( "-16g^2 + 192g = 288<BR>\n" );
document.write( "subtract 288 from both sides to get:<BR>\n" );
document.write( "-16g^2 + 192g - 288 = 0<BR>\n" );
document.write( "multiply both sides by -1 to get:<BR>\n" );
document.write( "16g^2 - 192g + 288 = 0<BR>\n" );
document.write( "divide both sides by 16 to get:<BR>\n" );
document.write( "g^2 - 12g + 18 = 0<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "use the quadratic formula to solve for g.<BR>\n" );
document.write( "a factor = 1<BR>\n" );
document.write( "b factor = -12<BR>\n" );
document.write( "c factor = 18<BR>\n" );
document.write( "quadratic formula is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28-b+%2B-+sqrt%28b%5E2-4ac%29%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "g becomes:<BR>\n" );
document.write( "(12 +/- sqrt(72))/2<BR>\n" );
document.write( "this turns out to be:<BR>\n" );
document.write( "g = 10.24264069<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "g = 1.757359313<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "using these values of g and solving for c in the equation of (8-c)*g = 24, we get:<BR>\n" );
document.write( "c = 5.656854249<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "c = -5.656854249<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since c is the speed of the current, it can't be negative, so the only good answer is c = 5.656854249.<BR>\n" );
document.write( "since this c was calculated with g = 1.757359313, that value of g is no good either.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we should have all we need:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we have:<BR>\n" );
document.write( "g = 10.24264069<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "c = 5.656854249<BR>\n" );
document.write( "and:<BR>\n" );
document.write( "f = 1.757359313 which was found by taking 12 - g since f + g equals 12.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the question was what is the speed of the current and the answer is:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "speed of the current is c which equals 5.656854249 miles per hour.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "to confirm the answer is ccorrect, we substitute in the original equations.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the original equation said that the boat went down stream for 24 miles and back up stream for 24 miles in a total of 12 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "f is the time it took to get downstream.<BR>\n" );
document.write( "g is the time it took to get back up stream<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "going downstream the boat went 8 miles per hour with a current of 5.656854249  miles per hour in the same direction for a total of 13.65685425 miles per hour taking a time of 1.757359313 hours to go a distance of 24 miles.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "coming back upstream, the boat went 8 miles per hour against a current of 5.656854249 miles per hour in the opposite direction for a total of 2.343145751 miles per hour taking a time of 10.24264069 hours to go a distance of 24 miles.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "total time taken was 1.757359313 hours plus 10.24264069 hours = 12 hours.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "all the numbers check out so the current speed is accurate and the answer is:<BR>\n" );
document.write( "speed of the current is 5.656854249 miles per hour.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );