document.write( "Question 703463: How to find point C on x-axis to get AC+BC is minimum with A(1,6) and B(6,1)? Thanks. \n" ); document.write( "
Algebra.Com's Answer #433635 by AnlytcPhil(1806)\"\" \"About 
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document.write( "Let the desired point on the x axis be C(x,0)\r\n" );
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document.write( "Using the distance formula:\r\n" );
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document.write( "D = AC+BC\r\n" );
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document.write( "D = \"sqrt%28%28x-1%29%5E2%2B%286-0%29%5E2%29%2Bsqrt%28%28x-6%29%5E2%2B%281-0%29%5E2%29\"\r\n" );
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document.write( "D = \"sqrt%28%28x-1%29%5E2%2B36%29%2Bsqrt%28%28x-6%29%5E2%2B1%29\"\r\n" );
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document.write( "We need to find \"dD%2Fdx\"\r\n" );
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document.write( "Both those terms on the right are of the form \r\n" );
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document.write( "u = \"sqrt%28v%5E2%2Bk%29\"\r\n" );
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document.write( "Differentiate that and you get\r\n" );
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document.write( "\"du%2Fdx\" = \"expr%28v%2Fsqrt%28v%5E2%2Bk%29%29%2Aexpr%28dv%2Fdx%29\"\r\n" );
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document.write( "So, using that on both terms: \r\n" );
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document.write( "\"dD%2Fdx\" = \"%28x-1%29%2Fsqrt%28%28x-1%29%5E2%2B36%29\" + \"%28x-6%29%2Fsqrt%28%28x-6%29%5E2%2B1%29\"\r\n" );
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document.write( "\"dD%2Fdx\" = \"%28x-1%29%2Fsqrt%28x%5E2-2x%2B1%2B36%29\" + \"%28x-6%29%2Fsqrt%28x%5E2-12x%2B36%2B1%29\"\r\n" );
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document.write( "\"dD%2Fdx\" = \"%28x-1%29%2Fsqrt%28x%5E2-2x%2B37%29\" + \"%28x-6%29%2Fsqrt%28x%5E2-12x%2B37%29\"\r\n" );
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document.write( "We set that = 0 to find minimum value:\r\n" );
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document.write( "\"%28x-1%29%2Fsqrt%28x%5E2-2x%2B37%29\" + \"%28x-6%29%2Fsqrt%28x%5E2-12x%2B37%29\" = 0\r\n" );
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document.write( "Clearing of fractions:\r\n" );
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document.write( "\"%28x-1%29sqrt%28x%5E2-12x%2B37%29\" + \"%28x-6%29sqrt%28x%5E2-2x%2B37%29\" = 0\r\n" );
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document.write( "\"%28x-1%29sqrt%28x%5E2-12x%2B37%29\" = \"-%28x-6%29sqrt%28x%5E2-2x%2B37%29\"\r\n" );
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document.write( "Square both sides\r\n" );
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document.write( "(x-1)²(x²-12x+37) = (x-6)²(x²-2x+37)\r\n" );
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document.write( "(x²-2x+1)(x²-12x+37) = (x²-12x+36)(x²-2x+37)\r\n" );
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document.write( "Multiply that out and combine terms and get:\r\n" );
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document.write( "x4-14x³+62x²-86x+37 =  x4-14x³+97x²-516x+1332\r\n" );
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document.write( "That simplifies to:\r\n" );
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document.write( "35x²-430x+1295 = 0\r\n" );
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document.write( "Divide thru by 5: \r\n" );
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document.write( "7x²-86x+259 = 0\r\n" );
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document.write( "That factors as\r\n" );
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document.write( "(7x-37)(x-7)=0\r\n" );
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document.write( "That has solutions x=\"37%2F7\" and x=7.\r\n" );
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document.write( "We can show that x=7 is extraneous, since both terms of\r\n" );
document.write( "\"dD%2Fdx\" are positive wnen x=7, so it isn't 0 there.\r\n" );
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document.write( "Thus x=\"37%2F7\" is the only solution to \"dD%2Fdx\" = 0 \r\n" );
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document.write( "x=\"37%2F7\" is between 1 and 6. It's easy to show that \r\n" );
document.write( "\"dD%2Fdx\" is negative when x=1 and positive when x=6, \r\n" );
document.write( "thus the desired point is C(\"37%2F7\",0).\r\n" );
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document.write( "The graph showing the minimum of AC+BC drawn to scale is:\r\n" );
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document.write( "Edwin
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