10.6 Applications

A producer of personal computer mouse covers determines that the number $N$ of covers sold is related to the price $x$ of a cover by $N=35x-x^2.$ At what price should the producer price a mouse cover in order to sell 216 of them?

$$\begin{array}{l}\begin{array}{ccc}\text{Step\hspace{0.17em}}1:& & \text{Let}x=\text{the}\text{price}\text{of}\text{a}\text{mouse}\text{cover}\text{.}\end{array}\hfill \\ \begin{array}{lll}\text{Step\hspace{0.17em}}2:\hfill & \hfill & \text{Since}N\text{is}\text{to}\text{be}\text{216,}\text{the}\text{equation}\text{is}\hfill \\ \hfill & \hfill & 216=35x-x^2\hfill \end{array}\hfill \\ \begin{array}{lllllll}\text{Step\hspace{0.17em}}3:\hfill & \hfill & 216\hfill & =\hfill & 35x-x^2\hfill & \hfill & \text{Rewrite}\text{in}\text{standard}\text{form}\text{.}\hfill \\ \hfill & \hfill & x^2-35x+216\hfill & =\hfill & 0\hfill & \hfill & \text{Try}\text{factoring}\text{.}\hfill \\ \hfill & \hfill & (x-8)(x-27)\hfill & =\hfill & 0\hfill & \hfill & \hfill \\ \hfill & \hfill & x-8=0\hfill & or\hfill & \hfill & x-27=0\hfill & \hfill \\ \hfill & \hfill & x=8\hfill & or\hfill & \hfill & x=27\hfill & \hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{Check}\text{these}\text{potential}\text{solutions}\text{.}\end{array}\hfill \\ \begin{array}{llllllllll}\text{Step\hspace{0.17em}}4:\hfill & \hfill & \text{If}x=8,\hfill & \hfill & \hfill & \hfill & \text{If}x=27,\hfill & \hfill & \hfill & \hfill \\ \hfill & \hfill & \hfill 35·8-8^2& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 35·27-{27}^2& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 280-64& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill & \hfill 945-729& =\hfill & 216\hfill & \text{Is\hspace{0.17em}this\hspace{0.17em}correct?}\hfill \\ \hfill & \hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill & \hfill 216& =\hfill & 216\hfill & \text{Yes,\hspace{0.17em}this\hspace{0.17em}is\hspace{0.17em}correct}\text{.}\hfill \end{array}\hfill \\ \begin{array}{ccc}& & \text{These}\text{solutions}\text{check}\text{.}\end{array}\hfill \\ \hfill \begin{array}{ccc}\text{Step}5:& & \text{The}\text{computer}\text{mouse}\text{covers}\text{can}\text{be}\text{priced}\text{at}\text{either}\text{\$8}\text{or}\text{\$27}\text{in}\text{order}\text{to}\text{sell}\text{216}\text{of}\text{them}\text{.}\end{array}\end{array}$$