document.write( "Question 1193026: What is the maximum product of 2 numbers whose sum is 45? What numbers yield this product? \n" ); document.write( "

Algebra.Com's Answer #824971 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "What is the maximum product of 2 numbers whose sum is 45? What numbers yield this product?
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document.write( "Let x and y be the numbers; let x be the lesser number and y be the greater number.\r\n" );
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document.write( "Consider the mean value a =  =  = 22.5  and half the distance between x and y,  b = \r\n" );
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document.write( "Then, obviously,  x = 22.5 - b  and  y = 22.5 + b, so we can write\r\n" );
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document.write( "    xy = (22.5-b)*(22.5+b) = .\r\n" );
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document.write( "From this formula, it is clear that xy is maximum when b= 0, i.e. when the numbers x and y are equal: x = y.\r\n" );
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document.write( "Thus we get that the maximum of the product xy  is achieved when x = y.\r\n" );
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document.write( "Together with x + y = 45 it implies that the maximum is achieved in this problem at x = y = 45/2 = 22.5.\r\n" );
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document.write( "The maximum value of xy is  then   = 506.25.   \r\n" );
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document.write( "ANSWER.  The maximum is achieved when both numbers are equal to half of the sum, i.e. 22.5.\r\n" );
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document.write( "         The maximum value of the product is  506.25  then.\r\n" );
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\n" ); document.write( "\n" ); document.write( "There are many different ways and methods of solving such problems.\r
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\n" ); document.write( "\n" ); document.write( "They differ in the level of necessary prerequisites, which the reader should know to understand the solution.\r
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\n" ); document.write( "\n" ); document.write( "What I exposed here, is traditionally considered in Math and in Math popular literature as \r
\n" ); document.write( "\n" ); document.write( "the SIMPLEST method of solving, accessible for students with minimal level of knowledge (for the most young readers).\r
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