Chapter 4.5, Problem 43ES

(a) A chemical manufacturer sells sulfuric acid in bulk at a price of $\text{\$}100$ per unit. If the daily total production cost in dollars for x units is $C\left(x\right)=100,000+50x+0.0025x^2$ and if the daily production capacity is at most $7000$ units, how many units of sulfuric acid must be manufactured and sold daily to maximize the profit?

(b) Would it benefit the manufacturer to expand the daily production capacity?

(c) Use marginal analysis to approximate the effect on profit if daily production could be increased from $7000$ to $7001$ units.