Find the quantity that produces maximum profit, given the demand function 
and cost function 
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| Foundations:
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Recall that the demand function,  , relates the price per unit Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p}
to the number of units sold, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x}
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Moreover, we have several important important functions:
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 , the total cost to produce  units;
 , the total revenue (or gross receipts) from producing units;
- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(x)}
, the total profit from producing units.
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| In particular, we have the relations
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| and
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| Using these equations, we can find the maximizing production level by determining when the first derivative of profit is zero.
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Solution:
| Step 1:
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| Find the Profit Function: We have
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle R(x)\,=\,x\cdot p(x)\,=\,x\cdot (90-3x)\,=\,90x-3x^{2}.}
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| From this,
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(x)\,=\,R(x)-C(x)\,=\,90x-3x^{2}-\left(200-30x+x^{2}\right)\,=\,120x-4x^{2}-200.}
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| Step 2:
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| Find the Maximum: The equation for marginal revenue is
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- Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle P(x)\,=\,120x-4x^{2}-200.}
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| Applying our power rule to each term, we find
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- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P'(x)\,=\,120-8x\,=\,8(15-x).}
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| The only root of this occurs at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=15}
, and this is our production level to achieve maximum profit.
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| Final Answer:
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| Maximum profit occurs when we produce 15 items.
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