document.write( "Question 243614: Find two positive numbers such the sum of the first and twice the second is 100 and their product is as large as possible? \n" ); document.write( "
Algebra.Com's Answer #178521 by Edwin McCravy(20054)\"\" \"About 
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Find two positive numbers such the sum of the first and twice the second is 100 and their product is as large as possible?
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document.write( "Let t = the first number\r\n" );
document.write( "Let x = the second number\r\n" );
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document.write( ">>...the sum of the first and twice the second is 100...<<\r\n" );
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document.write( "\"t%2B2x+=+100\"\r\n" );
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document.write( "We want to maximize the product xy.  \r\n" );
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document.write( "Let y = the product = tx\r\n" );
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document.write( "\"y=tx\"\r\n" );
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document.write( "Solve \"t%2B2x+=+100\" for t\r\n" );
document.write( "      \"t=+100-2x\"\r\n" );
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document.write( "Substitute \"100-2x\" for t in\r\n" );
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document.write( "\"y+=+tx\"\r\n" );
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document.write( "\"y+=+%28100-2x%29x\"\r\n" );
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document.write( "\"y+=+x%28100-2x%29\"\r\n" );
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document.write( "\"y+=+100x-2x%5E2\"\r\n" );
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document.write( "If you are taking algebra, not calculus, then find the\r\n" );
document.write( "vertex of the parabola\r\n" );
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document.write( "\"y=-2x%5E2%2B100x%2B0\"\r\n" );
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document.write( "Since the coefficient of \"x%5E2\" is negative, the\r\n" );
document.write( "graph opens downward and thus reaches a maximum at the\r\n" );
document.write( "vertex.\r\n" );
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document.write( "The formula for the x-coordinate of the vertex = \"-b%2F%282a%29\"\r\n" );
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document.write( "So the x-coordinate of the vertex is \"-b%2F%282a%29=-%28100%29%2F%282%28-2%29%29=100%2F4=25\"\r\n" );
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document.write( "So the value of x is 25\r\n" );
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document.write( "Now substitute 25 for x in \r\n" );
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document.write( "\"t%2B2x+=+100\"\r\n" );
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document.write( "\"t%2B2%2825%29+=+100\"\r\n" );
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document.write( "\"t%2B50+=+100\"\r\n" );
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document.write( "\"t=50\"\r\n" );
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document.write( "So the two numbers are\r\n" );
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document.write( "first = 25\r\n" );
document.write( "second = 50\r\n" );
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document.write( "---------------------------------------------\r\n" );
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document.write( "If you're taking calculus, you would do it by\r\n" );
document.write( "taking the derivative of \r\n" );
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document.write( "\"y+=+100x-2x%5E2\"\r\n" );
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document.write( "\"%28dy%29%2F%28dx%29=100-4x\"\r\n" );
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document.write( "and set the derivative = 0\r\n" );
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document.write( "\"100-4x=0\"\r\n" );
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document.write( "\"-4x=-100\"\r\n" );
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document.write( "\"x=25\"\r\n" );
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document.write( "and as in the algebra way, \"t=100-2x=100-2%2825%29=100-50=50\" \r\n" );
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document.write( "Edwin
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