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### Section 4.14 : Business Applications

5. The production costs, in dollars, per week of producing $$x$$ widgets is given by,

$C\left( x \right) = 4000 - 32x + 0.08{x^2} + 0.00006{x^3}$

and the demand function for the widgets is given by,

$p\left( x \right) = 250 + 0.02x - 0.001{x^2}$

What is the marginal cost, marginal revenue and marginal profit when $$x = 200$$ and $$x = 400$$? What do these numbers tell you about the cost, revenue and profit?

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First, we need to get the revenue and profit functions. From the notes for this section we know that these functions are,

\begin{align*}{\mbox{Revenue : }} & R\left( x \right) = x\,p\left( x \right) = 250x + 0.02{x^2} - 0.001{x^3}\\ {\mbox{Profit : }} & P\left( x \right) = R\left( x \right) - C\left( x \right) = - 4000 + 282x - 0.06{x^2} - 0.00106{x^3}\end{align*} Show Step 2

From the notes in this section we know that the marginal cost, marginal revenue and marginal profit functions are simply the derivative of the cost, revenue and profit functions so let’s start with those.

\begin{align*}C'\left( x \right) & = - 32 + 0.16x + 0.00018{x^2}\\ R'\left( x \right) & = 250 + 0.04x - 0.003{x^2}\\ P'\left( x \right) & = 282 - 0.12x - 0.00318{x^2}\end{align*} Show Step 3

The marginal cost, marginal revenue and marginal profit for each value of $$x$$ is then,

\require{bbox} \bbox[2pt,border:1px solid black]{\begin{align*}C'\left( {200} \right) & = 7.2 & \hspace{0.75in}R'\left( {200} \right) & = 138 & \hspace{0.5in}P'\left( {200} \right) & = 130.8\\ C'\left( {400} \right) & = 60.8 & \hspace{0.5in}R'\left( {400} \right) & = - 214 & \hspace{0.5in}P'\left( {400} \right) & = - 274.8\end{align*}} Show Step 4

From these computations we can see that producing the 201st widget will cost approximately $7.2 and will add approximately$138 in revenue and $130.8 in profit. Likewise, producing the 401st widget will cost approximately$60.8 and will see a decrease of approximately $214 in revenue and a decrease of$274.8 in profit.