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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 = = 1150;
hence, it reaches the maximum at = = 350
(precisely at the half-way between the roots -450 and 1150).
Thus the optimum production is 450 + 350 = 800 items at the price
= 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.
ANSWER. 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.
Solved.