document.write( "Question 1183764: Let (x,y) be a point on the triangular region bounded by the line 3x + 4y = 12 and the coordinate axes. Determine the points (x,y) in this region which give the minimum and maximum sums of distances of the point from the line 3x + 4y = 12 and from the coordinate axes. Please include explanation. Thank you! \n" ); document.write( "

Algebra.Com's Answer #814236 by ikleyn(52817) $\"\"$ $\"About$

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\n" ); document.write( "Let (x,y) be a point on the triangular region bounded by the line 3x + 4y = 12 and the coordinate axes.
\n" ); document.write( "Determine the points (x,y) in this region which give the minimum and maximum sums of distances of the point
\n" ); document.write( "from the line 3x + 4y = 12 and from the coordinate axes. Please include explanation. Thank you!
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document.write( "Make a sketch.\r\n" );
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document.write( "You have a right triangle ABC with the right angle at vertex A = (0,0); \r\n" );
document.write( "the point B is x-intercept (4,0) and the point C is y-intercept (0,3).\r\n" );
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document.write( "So the legs of the triangle are  AB = 4, BC = 3 and the hypotenuse BC is 5 units long.\r\n" );
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document.write( "Let P = (x,y) be the point inside triangle ABC (including its sides).\r\n" );
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document.write( "Then the distance from P to x-axis is y; the distance from P to y-axis is x.\r\n" );
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document.write( "To find the distance from P to the hypotenuse BC, consider triangles PAB, PAC and PBC.\r\n" );
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document.write( "We will find the distance from P to BC as the altitude of the triangle PBC.\r\n" );
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document.write( "The area of the triangle ABC is   =  = 6 square units;\r\n" );
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document.write( "the area of the triangle PAB is   =  = 2y;\r\n" );
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document.write( "the area of the triangle PBC is   =  = 1.5x.\r\n" );
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document.write( "The area of the triangle PAC,  ,  is the difference  -  -  = 6 - 2y - 1.5x.\r\n" );
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document.write( "The altitude of the triangle PBC drawn to BC is TWICE its area divided by the length of BC, i.e.   = 2.4 - 0.6x - 0.8y.\r\n" );
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document.write( "Thus, the sum of the distances from the point P(xy) to the sides of the triangle ABC is  x + y + 2.4 - 0.6x - 0.8y = 2.4 + 0.4x + 0.2y.\r\n" );
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document.write( "The problem asks to find the points P(x,y) inside the triangle ABC, which provide the minimum / the maximum to this function f(x,y) = 2.4 + 0.4x + 0.2y.\r\n" );
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document.write( "This function is linear function of x and y, and it is not a constant.\r\n" );
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document.write( "THEREFORE, the function gets its maximum/minimum inside the triangle at one of the vertices of the triangle.\r\n" );
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document.write( "So, I prepared the Table below, which shows the value of the function at the vertices of the triangle ABC\r\n" );
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document.write( "            T    A    B    L    E\r\n" );
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document.write( "                          f(x,y)\r\n" );
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document.write( "    Point A (0,0)      2.4 + 0.4*0 + 0.2*0 = 2.4\r\n" );
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document.write( "    Point B (4,0)      2.4 + 0.4*4 + 0.2*0 = 4.0\r\n" );
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document.write( "    Point C (0,3)      2.4 + 0.4*0 + 0.2*3 = 3.0\r\n" );
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document.write( "From the Table,  the sum of the distances is maximum at the vertex B(4,0).\r\n" );
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document.write( "                 The sum of the distances is minimum at the vertex A(0,0).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, answered and thoroughly explained.\r
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\n" ); document.write( "\n" ); document.write( "I just fixed my error after notice by @greenestamps, and now you see the fixed correct version.\r
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\n" ); document.write( "\n" ); document.write( "Thanks to @greenestamps !\r
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