document.write( "Question 1207302: The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b) where A and B are positive numbers. What is the value of a/b? \n" ); document.write( "

Algebra.Com's Answer #845095 by ikleyn(52792) $\"\"$ $\"About$

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\n" ); document.write( "The lengths of the sides of an equilateral triangle are log4(a), log10(b), log25(a+b)
\n" ); document.write( "where A and B are positive numbers. What is the value of a/b?
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document.write( "We are given \r\n" );
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document.write( "     =  = .\r\n" );
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document.write( "Let  k =  =  = .\r\n" );
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document.write( "It means that \r\n" );
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document.write( "     = a,       (1)\r\n" );
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document.write( "     = b,      (2)\r\n" );
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document.write( "     = a + b.  (3)\r\n" );
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document.write( "It implies that\r\n" );
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document.write( "     +  = .\r\n" );
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document.write( "Divide both sides (all the terms) by  .  You will get\r\n" );
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document.write( "     +  = 1,\r\n" );
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document.write( "or\r\n" );
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document.write( "     +  = 1.    (4)\r\n" );
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document.write( "Let x = .  Then equation (4) takes the form\r\n" );
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document.write( "     + x = 1,\r\n" );
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document.write( "or\r\n" );
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document.write( "     + x - 1 = 0.\r\n" );
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document.write( "Its roots are   = ,   = .\r\n" );
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document.write( "Our value of x is positive ;  so, we consider only positive root  x = .\r\n" );
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document.write( "Thus we have\r\n" );
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document.write( "     = ,\r\n" );
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document.write( "or\r\n" );
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document.write( "     = .\r\n" );
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document.write( "But from (1) and (2),   = .\r\n" );
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document.write( "Thus we proved that   = .\r\n" );
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document.write( "ANSWER.   =  = 0.618033989  (approximately).\r\n" );
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