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.