Example 6.3.16 Profit, Revenue, and Costs

From Example 6.3.13, we know Bayani's 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*}