document.write( "Question 1192718: The car is moving south at a constant speed of 40km/h. The second car was initially located 60 km west of the first car, moving southeast at 50 km/h. What is the minimum distance between these two cars? \n" ); document.write( "

Algebra.Com's Answer #848816 by ikleyn(52794) $\"\"$ $\"About$

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\n" ); document.write( "The car is moving south at a constant speed of 40km/h.
\n" ); document.write( "The second car was initially located 60 km west of the first car, moving southeast
\n" ); document.write( "at 50 km/h. What is the minimum distance between these two cars?
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document.write( "Imagine a coordinate plane in your mind.\r\n" );
document.write( "The origin of this coordinate system is the point (0,0).\r\n" );
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document.write( "The original coordinate of the first car is (0,0).\r\n" );
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document.write( "Its current position at the time moment \"t\" is  (0,-40t), in kilometers\r\n" );
document.write( "where t is in hours.\r\n" );
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document.write( "The original coordinate of the second car is (-60,0).\r\n" );
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document.write( "Its current position at the time moment \"t\" is  (,), in kilometers\r\n" );
document.write( "where t is in hours.\r\n" );
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document.write( "The difference in x-coordinates is dx = .\r\n" );
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document.write( "The difference in y-coordinates is dy = .\r\n" );
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document.write( "The square of the distance is  \r\n" );
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document.write( "     = (dx)^2 + (dy)^2 =  + \r\n" );
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document.write( "        = 3600 -  +  + 1600*t^2 -  +  = \r\n" );
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document.write( "        = 3600 -  +  = 3600 -  + .    (1)\r\n" );
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document.write( "d is minimal where the quadratic form  is minimal.\r\n" );
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document.write( "The quadratic form    is minimal at  t = ,  where b is the coefficient of the form  at t,\r\n" );
document.write( "while \"a\" is the coefficient of the form d^2 at t^2.  So,\r\n" );
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document.write( "     =  =  = 1.669 hours (approximately).\r\n" );
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document.write( "To find the value of minimum d,  substitute t = 1.669  hours into the formula (1).  You will get\r\n" );
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document.write( "     =  = 61.0765.\r\n" );
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document.write( "Now the last step to find the value of the minimum distance\r\n" );
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document.write( "      =  =  = 7.815 km.\r\n" );
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document.write( "ANSWER.  The minimum distance between the cars is 7.815 km  (approximately)\r\n" );
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document.write( "         at the time moment t =  hours = 1.669  hours  (approximately).\r\n" );
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\n" ); document.write( "\n" ); document.write( "On finding the maximum/minimum of a quadratic function see the lessons\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the minimum/maximum of a quadratic function\r
\n" ); document.write( "\n" ); document.write( "    - HOW TO complete the square to find the vertex of a parabola\r
\n" ); document.write( "\n" ); document.write( "    - Briefly on finding the vertex of a parabola\r
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\n" ); document.write( "\n" ); document.write( "Consider these lessons as your textbook, handbook, tutorials and (free of charge) home teacher.\r
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\n" ); document.write( "\n" ); document.write( "It is a standard information and a standard piece of knowledge necessary for finding minimum/maximum of quadratic forms.\r
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