Section 4.14 : Business Applications
5. The production costs, in dollars, per week of producing x widgets is given by,
C(x)=4000−32x+0.08x2+0.00006x3and the demand function for the widgets is given by,
p(x)=250+0.02x−0.001x2What 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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Start SolutionFirst, we need to get the revenue and profit functions. From the notes for this section we know that these functions are,
Revenue : R(x)=xp(x)=250x+0.02x2−0.001x3Profit : P(x)=R(x)−C(x)=−4000+282x−0.06x2−0.00106x3 Show Step 2From 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.
C′(x)=−32+0.16x+0.00018x2R′(x)=250+0.04x−0.003x2P′(x)=282−0.12x−0.00318x2 Show Step 3The 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 4From 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.