document.write( "Question 1194697: From a car traveling east at 40 mi.per hr., an airplane traveling horizontally
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Algebra.Com's Answer #826941 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "We place the initial position of the car at the origin of the coordinate system (x,y,z) = (0,0,0).\r\n" );
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document.write( "Then the airplane's initial position is (x,y,z) = (1,-2,2).\r\n" );
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document.write( "The trajectory of the car   in time is  (40t,0,0).\r\n" );
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document.write( "The trajectory of the plane in time is  (0,-2+100t,2).\r\n" );
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document.write( "The vector from the car to the airplane in the coordinate form is  (-40t,-2+100t,2).\r\n" );
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document.write( "The square of the length of this vector (= the distance between the objects) is\r\n" );
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document.write( "    d^2(t) =  =  = \r\n" );
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document.write( "                 = .\r\n" );
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document.write( "The distance d(t) is minimum when d^2(t) is minimum.\r\n" );
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document.write( "d^2(t) is minimum at  t = \"  \" = - = 0.017241 of an hour = 1.0345 minutes.\r\n" );
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document.write( "At this time moment, the square of the distance is  d^2(t) =  = 8.0069 mi^2,\r\n" );
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document.write( "hence,  the distance itself is  d(t) =  = 2.830 miles.\r\n" );
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document.write( "Compare it with the initial distance  d(0) =  =  = 3 miles.\r\n" );
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\n" ); document.write( "\n" ); document.write( "On finding 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
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