# How do I interpret this mathematical model?

"A pharmaceutical company firm manufactures a liquid form of penicillin in bulk at a price of £200 per unit. The total production cost £$C$ for $x$ units is:

$$C = 500,000 + 80x + 0.003x^2$$

and the production capacity of the firm is at most 30,000 units in a specified time.

(a) Write down the revenue £$R$ the firm makes by selling $x$ units of penicillin."

I assume the profit made on each unit is 80$x$, but apart from that I'm not sure how I should go about interpreting and answering this. Thanks in advance.

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The profit is not $80x$ but the difference between the sales receipts and the production costs.

It seems that the sales receipts for $x$ units are $£200x$ and the cost is $£C$. The revenue is probably meant to be the difference between these two, assuming all items produced are sold. You are probably expected to state this as a function of $x$.

You can then have the pleasure of finding the minimum number of units where net revenues are positive (about 4725) and the number of units where net revenues are maximised (about 20000).

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Thanks, that clears a few things up for me. The only other thing I'm really struggling with here is my method, how would I go about answering the question? – Christy Mcgrory Jan 9 '12 at 2:00

Revenue is how much you make, full stop, ignoring the costs of production. So in this case, look at how much the company makes by selling $x$ units of penicillin.

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Costs: $$C=500000+80x+0.03x^2$$ Income: $$I=200x$$ Profit/Revenue: $$R=I-C=200x-(500000+80x+0.03x^2)=-0.03x^2+120x-500000$$

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