Question 1197728
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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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If the number of produced items is n more than 450, then the price per each item is 46-0.04 dollars.

Hence, the profit is then  P(n) = (450+n)*(46-0.04n) dollars.


P(n) in this form is a quadratic function; it has the zeroes at 

     n= -450 and n = {{{46/0.04}}} = 1150;


hence, it reaches the maximum at  {{{n[max]}}} = {{{((-450)+1150)/2}}} = 350
(precisely at the half-way between the roots -450 and 1150).



Thus the optimum  production is 450 + 350 = 800 items at the price  

    {{{46.00 - 0.04*350}}} = 32 dollars per item.


It gives the maximum profit of  800*32 = 25600 dollars.



For completeness purposes, compare it with the original profit of 450*46 = 20700 dollars for 460 items.


<U>ANSWER</U>.  At given conditions, the optimum number of produced items is 800 per week.

         It provides the maximum profit of 25600 dollars, against 20700 dollars at the original conditions.
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Solved.