College Algebra, Chapter 1, 1.3, Section 1.3, Problem 86

The formula $\displaystyle P = \frac{x}{10} (300 - x)$ represents the profit (in dollars) generated by producing $x$ microwave ovens per week, where $0 \leq x \leq 200$. How many ovens must be manufactured in a given week to generate a profit of $\$ 1250$?


$\begin{equation} \begin{aligned} P =& \frac{x}{10} (300 - x) && \text{Model} \\ \\ 1250 =& \frac{x}{10} (300 - x) && \text{Substitute the given} \\ \\ 1250 =& 30x - \frac{x^2}{10} && \text{Apply Distributive Property} \\ \\ 12500 =& 300x - x^2 && \text{Multiply both sides by } 10 \\ \\ x^2 - 300x + 12500 =& 0 && \text{Add $x^2$ and subtract } 300x \\ \\ (x - 250)(x - 50) =& 0 && \text{Factor} \\ \\ x - 250 =& 0 \text{ and } x - 50 = 0 && \text{ZPP} \\ \\ x =& 250 \text{ and } x = 50 && \text{Solve for } x \\ \\ x =& 50 && \text{Choose } x = 50, \text{ since } 0 \leq x \leq 200 \end{aligned} \end{equation}$


In order to generate a profit of $\$ 1250$, $50$ ovens must be manufactured.