Chapter 2.7, Problem 16E

To determine

To find: the profit if plant processing is maximized; if not possible state whether the problem is infeasible, unbounded or has alternate optimal solutions.

Answer to Problem 16E

$23250

Explanation of Solution

Given information:

Let the aluminum container for food be $x$ and container for drinks be $y$ .

Maximum aluminum processed per week = 1200 tons

Minimum aluminum processed for food container = 300 tons

Minimum aluminum processed for drink container = 450 tons

Profit on food container per ton = $17.50

Profit on drink container per ton = $20

Calculation:

Let $P\left(x,y\right)$ be the maximum profit function. The maximum profit is $P\left(x,y\right)=17.5x+20y$ .

The inequalities derived from the given information are as follows:

  $\begin{array}{c}x\ge 300\text{ tons}\\ y\ge 450\text{ tons}\\ x+y\le 1200\text{ tons}\end{array}$

Use a graphing calculator to plot the inequalities as shown below.

  

The vertices are $\left(300,450\right),\left(300,900\right)\text{ and }\left(750,450\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}P\left(300,450\right)=17.5\left(300\right)+20\left(450\right)\\ =\$14250\\ P\left(300,900\right)=17.5\left(300\right)+20\left(900\right)\\ =\$23250\\ P\left(750,450\right)=17.5\left(750\right)+20\left(450\right)\\ =\$22125\end{array}$

Thus, the profit if plant maximized processing is $23250.