Question:medium

A liquid form of penicillin manufactured by a pharmaceutical company is sold in bulk at a price of $\$200$ per unit. If the total production cost (in dollars) for $x$ units is \[ C(x)=500{,}000+80x+0.003x^2, \] and the production capacity of the firm is at most $30{,}000$ units in a specified time, then how many units of penicillin must be manufactured and sold during that specified time to maximize the profit?

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Profit is maximized when Marginal Revenue ($MR$) equals Marginal Cost ($MC$):
- $MR = \frac{d}{dx}(200x) = 200$
- $MC = \frac{d}{dx}(C(x)) = 80 + 0.006x$
Set $MR = MC \implies 200 = 80 + 0.006x \implies 120 = 0.006x \implies x = 20,000$.
This is a standard and fast economic approach.