Example6.2.14Profit, Revenue, and Costs

From Example 6.2.11, we know the ketchup company's production costs, \(C\) (in thousands of dollars), for producing \(x\) thousand jars of ketchup is modeled by \(C=0.15x^2+6x+30\text{.}\) The revenue, \(R\) (in thousands of dollars), from selling the ketchup can be modeled by \(R=13x\text{,}\) where \(x\) stands for the number of thousands of jars of ketchup sold. The company's net profit can be calculated using the concept:

\begin{equation*} \text{net profit} = \text{revenue} - \text{costs} \end{equation*}

Assuming all products produced will be sold, a polynomial to model the company's net profit, \(P\) (in thousands of dollars) is:

\begin{align*} P \amp= R-C\\ \amp= \left(13x\right)-\left(0.15x^2+6x+30\right)\\ \amp= 13x-0.15x^2-6x-30\\ \amp= -0.15x^2+\left(13x+(-6x)\right)-30\\ \amp=-0.15x^2+7x-30 \end{align*}