Chapter 2.2, Problem 58E

Maximum Profit The cost per unit in the production of an MP3 player is $\text{\$}60$. The manufacturer charges $\text{\$}90$ per unit for orders of $100$ or less. To encourage large orders, the manufacturer reduces the charge by $\text{\$}0.15$ per MP3 player for each unit ordered in excess of $100$ (for example, the charge is reduced to $\text{\$}87$ per MP3 player for an order size of $120$ ).

(a) The table shows the profits $P$ (in dollars) for various numbers of units ordered, $x$. Use the table to estimate the maximum profit.

(b) Plot the points $\left(x,P\right)$ from the table in part (a). Does the relation defined by the ordered pairs represent $P$ as a function of $x$ ?

(c) Given that $P$ is a function of $x$, write the function and determine its domain. (Note: $P=R-C,$ where $R$ is revenue and $C$ is cost.)