document.write( "Question 1032574: As Kyle walked along a street at a constant speed, he noticed that every 12 minutes, a bus passed him traveling in the same direction, and every 4 minutes, a bus passed him traveling in the opposite direction as Kyle. If all of the buses travel at a constant speed and leave the terminals at each end of the street at equally spaced intervals, how many minutes long is that interval? \n" ); document.write( "

Algebra.Com's Answer #647508 by ikleyn(52797) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "As Kyle walked along a street at a constant speed, he noticed that every 12 minutes, a bus passed him traveling in the same
\n" ); document.write( "direction, and every 4 minutes, a bus passed him traveling in the opposite direction as Kyle. If all of the buses travel
\n" ); document.write( "at a constant speed and leave the terminals at each end of the street at equally spaced intervals, how many minutes
\n" ); document.write( "long is that interval?
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "0.  If all buses travel at a constant speed and leave the terminals at each end of the street at equally spaced intervals, \r\n" );
document.write( "    then the distance between each pair of two consecutive buses is a constant value. \r\n" );
document.write( "\r\n" );
document.write( "    Let L be the distance  between any two consecutive bases (measured in meters, for example).\r\n" );
document.write( "\r\n" );
document.write( "    The value L is the same for buses moving in the same direction as Kyle moves.\r\n" );
document.write( "    Also, it is the same value L for all buses moving in opposite direction, as well.\r\n" );
document.write( "\r\n" );
document.write( "    Also, let \"u\" be the speed of the buses (common for all bases) and \"v\" be the Kyle's walking speed. \r\n" );
document.write( "    Surely, each speed is relative to the ground, and we use the derived unit  for these speeds.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "1.  Now, let us consider (imagine) walking Kyle on the street and buses that move opposite to him.\r\n" );
document.write( "    Let us consider the very moment  when one of the buses on the street comes up to the same position as Kyle in his walk-way. \r\n" );
document.write( "    So, at the moment  this bus is near Kyle; Kyle moves ahead with his walking speed \"v\" and the next bus, which is now at the \r\n" );
document.write( "    distance L from Kyle, moves toward Kyle with the speed \"u\". We are given that the next bus will come up to the Kyle new position \r\n" );
document.write( "\r\n" );
document.write( "    in 4 minutes. Hence, we can write the equation for these two bodies, Kyle and the bus,  moving toward each other as\r\n" );
document.write( "\r\n" );
document.write( "       L = u*4 + v^4,   or   L = (u + v)*4,   or  u + v = .   (1).\r\n" );
document.write( "\r\n" );
document.write( "    This is a standard \"Travel and distance equation\" for two bodies moving uniformly toward each other. \r\n" );
document.write( "    Indeed, their relative speed (rate) is u+v. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  Next, let us consider (imagine) walking Kyle on the street and buses that move along the street in the same direction.\r\n" );
document.write( "\r\n" );
document.write( "    Let us consider the very moment  when one of the buses on the street comes up to the same position as Kyle in his walk-way. \r\n" );
document.write( "    So, at the moment  this bus is near Kyle; Kyle moves ahead with his walking speed \"v\" and the next bus, which is now at the \r\n" );
document.write( "    distance L behind Kyle, moves in the same direction as Kyle with the speed \"u\". We are given that the next bus will come up \r\n" );
document.write( "\r\n" );
document.write( "    to the Kyle new position in 12 minutes. Hence, we can write the equation for these two bodies moving in the same direction as\r\n" );
document.write( "\r\n" );
document.write( "       L = u*12 - v^12,   or   L = (u - v)*12,   or  u - v = .   (2).\r\n" );
document.write( "\r\n" );
document.write( "    This is a standard \"Travel and distance equation\" for two bodies moving uniformly in the same direction. \r\n" );
document.write( "    Indeed, their relative speed (rate) is u-v. \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3.  OK. Now we are at the finish line, as I always say in similar situations.\r\n" );
document.write( "\r\n" );
document.write( "    We have two equations (1) and (2) for velocities \"u\" and \"v\". Let us write them one more time as a system:\r\n" );
document.write( "\r\n" );
document.write( "        u + v = ,    (1')   and\r\n" );
document.write( "        u - v = .    (2')\r\n" );
document.write( "\r\n" );
document.write( "    Add equations (1') and (2') (both sides). You will get\r\n" );
document.write( "\r\n" );
document.write( "        2u =  + ,   or   2u =  =  = .\r\n" );
document.write( "\r\n" );
document.write( "    Hence,  u = .\r\n" );
document.write( "\r\n" );
document.write( "    What does it mean?\r\n" );
document.write( "    But of course, it means that the time the buses cover the distance L is  :  = 6 minutes.\r\n" );
document.write( "\r\n" );
document.write( "    It gives the answer to the problem:  the time interval buses leave their terminals is 6 minutes.\r\n" );
document.write( "

\n" ); document.write( " \n" ); document.write( "

\n" );