document.write( "Question 1197728: A manufacturer can make a profit of $46 on each item if not more than 450 items are produced each week. The profit on each item decreases 4 cents for every item over 450. To maximize profit, how many items should be produced each week?
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Algebra.Com's Answer #831460 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "A manufacturer can make a profit of $46 on each item if not more than 450 items
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document.write( "If the number of produced items is n more than 450, then the price per each item is 46-0.04 dollars.\r\n" );
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document.write( "Hence, the profit is then  P(n) = (450+n)*(46-0.04n) dollars.\r\n" );
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document.write( "P(n) in this form is a quadratic function; it has the zeroes at \r\n" );
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document.write( "     n= -450 and n =  = 1150;\r\n" );
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document.write( "hence, it reaches the maximum at   =  = 350\r\n" );
document.write( "(precisely at the half-way between the roots -450 and 1150).\r\n" );
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document.write( "Thus the optimum  production is 450 + 350 = 800 items at the price  \r\n" );
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document.write( "     = 32 dollars per item.\r\n" );
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document.write( "It gives the maximum profit of  800*32 = 25600 dollars.\r\n" );
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document.write( "For completeness purposes, compare it with the original profit of 450*46 = 20700 dollars for 460 items.\r\n" );
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document.write( "ANSWER.  At given conditions, the optimum number of produced items is 800 per week.\r\n" );
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document.write( "         It provides the maximum profit of 25600 dollars, against 20700 dollars at the original conditions.\r\n" );
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