document.write( "Question 1183317: At 3:00 PM, Andy began walking directly south from point A at a rate of 4 mi/h. At the same time, Bill started walking directly west from point B to point A, at a rate of 5 mi/h. The distance between A and B is 9 miles. Assume that both of them stop walking when Bill reaches point A. \r
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Algebra.Com's Answer #813563 by ikleyn(52817) $\"\"$ $\"About$

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\n" ); document.write( "At 3:00 PM, Andy began walking directly south from point A at a rate of 4 mi/h.
\n" ); document.write( "At the same time, Bill started walking directly west from point B to point A, at a rate of 5 mi/h.
\n" ); document.write( "The distance between A and B is 9 miles. Assume that both of them stop walking when Bill reaches point A.
\n" ); document.write( "At what time will be distance between Andy and Bill be minimized?
\n" ); document.write( "Round your answer to the nearest minute.
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document.write( "Place the origin of the coordinate system at the point A.\r\n" );
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document.write( "Then the point B has coordinates (9,0), from the context.\r\n" );
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document.write( "At time \"t\" hours after 3:00 PM, Andy's location is the point (0,-4t),\r\n" );
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document.write( "while Bill's location is the point (9-5t,0).\r\n" );
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document.write( "The square of the distance between Andy and Bill, as the function of time, is \r\n" );
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document.write( "    d^2(t) =  +  =  +  = .\r\n" );
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document.write( "The distance between the persons is minimal when the SQUARE of the distance is minimal.\r\n" );
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document.write( "Thus, to find the time of the minimal distance between the persons, it is enough to differentiate the function d^2(t) \r\n" );
document.write( "over t and equate it to zero.  It gives the equation\r\n" );
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document.write( "    2*41*t - 90 = 0,  or  82t = 90,    t =  hours = 1.097561 hours = 1 hour and 6 minutes (rounded to the nearest minute, as requested).\r\n" );
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document.write( "ANSWER.  The distance between the persons is minimal at t = 1 hour and 6 minutes after 3:00 PM, i.e. at 4:06 PM.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Below are the plots of the square of the distance $\"d%5E2%28t%29\"$ and the distance itself $\"d%28t%29\"$ as functions of time \"t\".\r
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document.write( "     Plots   d^2(t) =    (red)   and   d(t) =     (green)\r\n" );
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