Chapter 4.1, Problem 49E

OPTIMIZING PROFIT FOR COLD FORMULAS Beyer Pharmaceutical produces three kinds of cold formulas: I, II, and III. It takes $2.5\text{hr}$ to produce $1000$ bottles of Formula I, $3\text{hr}$ to produce $1000$ bottles of Formula II, and $4\text{hr}$ to produce $1000$ bottles of Formula III. The profits for each $1000$ bottles of Formula I, Formula II, and Formula III are $\$180$, $\$200$, and $\$300$, respectively. Suppose that for a certain production run, there are enough ingredients on hand to make at most $9000$ bottles of Formula I, $12,000$ bottles of Formula II, and $6000$ bottles of Formula III. Furthermore, suppose the time for the production run is limited to a maximum of $70\text{hr}$. How many bottles of each formula should be produced in this production run so that the profit is maximized? What is the maximum profit realizable by the company? Are there any resources left over?