document.write( "Question 1163673: Sam drives from City A to City B, and Richard drives from City B to City A on the same road. When they meet, Sam has driven 120 km. They continue to drive to their own destination. When they arrive at their destination, they turn around and continue to drive. When they meet the second time, the place is 1/5 of the distance between the two cities away from City B. What is the distance between the two cities? \n" ); document.write( "

Algebra.Com's Answer #787854 by ikleyn(52778) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Sam drives from City A to City B, and Richard drives from City B to City A on the same road. When they meet, Sam has driven 120 km.
\n" ); document.write( "They continue to drive to their own destination. When they arrive at their destination, they turn around and continue to drive.
\n" ); document.write( "When they meet the second time, the place is 1/5 of the distance between the two cities away from City B.
\n" ); document.write( "What is the distance between the two cities?\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Solution\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "The scheme of placing points is shown in the Figure below.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    +--------------------------------|-----------|--------------+\r\n" );
document.write( "\r\n" );
document.write( "    A                                M           F              B \r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Points A and B are cities;  point M is the 1st meeting point;  point F is the 2-nd meeting point.\r\n" );
document.write( "\r\n" );
document.write( "So, AM = 120 kilometers.\r\n" );
document.write( "\r\n" );
document.write( "Let x be the distance from the 1st meeting point to B.\r\n" );
document.write( "\r\n" );
document.write( "Thus the total distance is 120+x kilometers.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The times driving from the starting points to the 1st meeting pojnt are the same\r\n" );
document.write( "\r\n" );
document.write( "     = ,     (1)\r\n" );
document.write( "\r\n" );
document.write( "where    and   are the rates of Sam and Richard, respective.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "It is our first equation, which I want to rewrite in this form as the ratio of their rates\r\n" );
document.write( "\r\n" );
document.write( "     = .     (2)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The distance Sam     drove from the 1st meeting point to the second meeting point was   km.\r\n" );
document.write( "\r\n" );
document.write( "The time Sam     spent covering this distance was   =   hours.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The distance Richard drove from the 1st meeting point to the second meeting point was   km.\r\n" );
document.write( "\r\n" );
document.write( "The time Richard spent covering this distance was   =   hours.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The times    and   are the same;  it leads to equation\r\n" );
document.write( "\r\n" );
document.write( "   \r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In the right side, replace    by  ,  based on (2).  You will get then\r\n" );
document.write( "\r\n" );
document.write( "     = .\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Simplify it step by step\r\n" );
document.write( "\r\n" );
document.write( "     = \r\n" );
document.write( "\r\n" );
document.write( "     = \r\n" );
document.write( "\r\n" );
document.write( "    x*(3x + 60) = 120*(540+2x)\r\n" );
document.write( "\r\n" );
document.write( "    3x^2 + 60x = 120*540 + 240x \r\n" );
document.write( "\r\n" );
document.write( "    3x^2 - 180x - 120*540 = 0\r\n" );
document.write( "\r\n" );
document.write( "      x^2 - 60x - 21600 = 0    \r\n" );
document.write( "\r\n" );
document.write( "      (x+120)*(x-180) = 0\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Of the two roots,  -120 and 180,  only positive 180  makes sense.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "So, the distance x is 180 kilometers, and the total distance from A to B is  120+180 = 300 kilometers.     ANSWER\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Solved.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "---------------\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Surely, this problem is two levels higher than a regular Math/Physics problem in any high school.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "It is, actually, an Olympiad level problem in Physics.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Could you tell me please where is it from, from which source ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Do not forget to post your \"THANKS\" to me for my teaching.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );