document.write( "Question 1065804: As a rower passes under a bridge, a bottle of whiskey falls into the water.
\n" ); document.write( "Since it's half-full, it floats.
\n" ); document.write( "The rower doesn't hear it, and continues downstream.
\n" ); document.write( "After 20 minutes, he gets thirsty and looks for the bottle.
\n" ); document.write( "Having sobered some, he figures out that it fell into the water.
\n" ); document.write( "He turns around and rows upstream.
\n" ); document.write( "He finds the bottle 1 mile from the bridge.
\n" ); document.write( "***************
\n" ); document.write( "Find the speed of the current. \n" ); document.write( "
Algebra.Com's Answer #680994 by ikleyn(52797)\"\" \"About 
You can put this solution on YOUR website!
.
\n" ); document.write( "When I read this problem condition yesterday morning, automatic switcher worked in my mind/(undermind), and another/different
\n" ); document.write( "version came to my conscionsness:\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "    A rower goes upstream on a river.  When passing under a bridge,  a bottle of whiskey falls into the water.\r\n" );
document.write( "    Since it's half-full,  it floats.  The rower doesn't notice it,  and continues going upstream.\r\n" );
document.write( "    After  20 minutes,  he gets thirsty and looks for the bottle.  Having sobered some,  he figures out that it fell into the water.\r\n" );
document.write( "    He turns around and rows downstream.  He finds the bottle  1 mile from the bridge.  Find the speed of the current.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "Yesterday in the evening I got understanding that I actually solved DIFFERENT problem than the original.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "But thinking about the subject, I then got understanding that this SECOND formulation is actually BETTER than the original,
\n" ); document.write( "and is much more NATURAL.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Why ?? - If you read attentively both formulations, you will get \"why\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The second formulation, which came from \"my undermind\" is much more natural from the point of view of human motivation than the original.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Indeed, WHO with the healthy mind will start rowing against the current to get the bottle ??
\n" ); document.write( "Much simpler is to stay \"in place\" and to wait when the current will bring/deliver the bottle.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So, now I think (and I am 120% sure) that my undermined formulation is much better, is much more realistic and is much more similar/closer
\n" ); document.write( "to the classic, than that what came first.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So, my solution below and everything what follows relates to the UPDATED formulation, and NOT TO THE ORIGINAL ONE.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Sorry that this new understanding came with a delay. But later is better than no-when.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\n" ); document.write( "\r\n" ); document.write( " \r\n" ); document.write( " \r\n" ); document.write( " \r\n" ); document.write( "\n" ); document.write( "

\n" ); document.write( "    I checked my sources, and now my statement is:    \r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    My changed formulation IS CLASSICAL.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "    That which came originally, IS NOT.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r
\n" ); document.write( "\n" ); document.write( "/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This problem is classic Travel and Distance problem. (with the corrections above !!)
\n" ); document.write( "Edwin provided its classic solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "But his solution requires a powerful imagination.
\n" ); document.write( "For those who experiences a lack of such imagination (as myself), I will give another solution.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "
\r\n" );
document.write( "Let \"u\" be the rower speed in \"still water\" and \"v\" be the current speed (in miles per hour). \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "From that moment, when the rower figured out that the bottle fell into water,\r\n" );
document.write( "from the same moment when he turned around and start catching the bottle, we have a classic \"catching up\" problem.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Namely, we have the rower at the distance of \"%28u-v%29%2A%281%2F3%29\" upstream the bridge, and we have a bottle at the distance \"v%2A%281%2F3%29\" downstream the bridge.\r\n" );
document.write( "\r\n" );
document.write( "What is \"1%2F3\" here ? - But of course, it is \"1%2F3\" of an hour, or 20 minutes.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this moment we turn ON our chronometer, we start counting the time, and what we see ? What do we observe ?\r\n" );
document.write( "\r\n" );
document.write( "Starting from this moment and till the moment \"t\" of catching up the bottle, the rower will cover the distance \r\n" );
document.write( "\r\n" );
document.write( "D = (the distance back to the bridge) + (the distance the bottle went downstream in \"1%2F3\" of an hour) + (the distance bottle went downstream during the time \"t\") = \r\n" );
document.write( "\r\n" );
document.write( "         \"%28u-v%29%2A%281%2F3%29\"                     +     \"v%2A%281%2F3%29\"                                                       +     \"v%2At\" = \r\n" );
document.write( "\r\n" );
document.write( "       = \"u%2A%281%2F3%29+-+v%2A%281%2F3%29+%2B+v%2A%281%2F3%29\" + \"v%2At\" = \"u%2A%281%2F3%29\" + \"v%2At\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Calculating this distance by another way (using the speed (u+v) of the rower downstream) we can express this distance as\r\n" );
document.write( "\r\n" );
document.write( "D = \"%28u%2Bv%29%2At\".\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The distance is the same (!!!), so we have an equation \r\n" );
document.write( "\r\n" );
document.write( "\"u%2A%281%2F3%29\" + \"v%2At\" = \"%28u%2Bv%29%2At\".\r\n" );
document.write( "\r\n" );
document.write( "Simplify it in one more step, and you get A REMARKABLE EQUALITY:\r\n" );
document.write( "\r\n" );
document.write( "         t = \"1%2F3\".\r\n" );
document.write( "\r\n" );
document.write( "Which means (and we PROVED it) that the catching time is exactly \"1%2F3\" of an hour.\r\n" );
document.write( "\r\n" );
document.write( "The thing which was not obvious from the beginning !\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Now we are at the finish line. One step to the end. To the VICTORY !!\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "We have another condition: for 20 minutes plus \"t\" minutes the bottle traveled exactly one mile !\r\n" );
document.write( "\r\n" );
document.write( "It means that \"%281%2F3%29%2Av+%2B+%281%2F3%29%2Av\" = 1,  or  \"%282%2F3%29%2Av\" = 1,  or  (AT LAST !!!)  v = \"3%2F2\" miles per hour.\r\n" );
document.write( "
\r
\n" ); document.write( "\n" ); document.write( "    SOLVED.     SOLVED.     SOLVED. \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------------
\n" ); document.write( "I am a mathematician by my basic education, so my mind works automatically in attempting to reverse / to transform the nonsense to
\n" ); document.write( "some sensible. So it was in this case (as in many other cases).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "''''''''''''''''''''''''''''
\n" ); document.write( "Now I have a question to the person who posted (or invented ?) the original post:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Why (or for what reason) do you distribute this original the half-false formulation of the beautiful problem
\n" ); document.write( "carefully designed hundreds years ago by those who really know and love MATH ???\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The problems like this one (corrected by me) are/is a treasure, literally.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "And they require an adequate treatment, too. \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
\n" );