Chapter 2.7, Problem 19E

To determine

To find: the number of snack size and family size bags that must be processed in order to maximize profit; if not possible state whether the problem is infeasible, unbounded or has alternate optimal solutions.

Answer to Problem 19E

Snack size bags: 600

Family size bags: 1800

Explanation of Solution

Given information:

Let the snack size and family size bag be $x$ and $y$ respectively.

Total unit production of $x\text{ and }y$ per week = 2400

Minimum unit production of $x$ = 600

Minimum unit production of $y$ = 900

Profit ona unit of $x$ = $12

Profit on a unit of $y$ = $18

Calculation:

Let $P\left(x,y\right)$ be the maximum profit function. The maximum profit is as follows:

  $P\left(x,y\right)=12x+18y$ .

The inequalities derived from the given information are as follows:

  $\begin{array}{c}x\ge 600\\ y\ge 900\\ x+y\le 2400\end{array}$

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

  

The vertices are $\left(600,900\right)\text{,}\left(1500,900\right)\text{ and }\left(600,1800\right)$ . Now test each vertex as shown below.

  $\begin{array}{c}P\left(600,900\right)=12\left(600\right)+18\left(900\right)\\ =\$23400\\ P\left(1500,900\right)=12\left(1500\right)+18\left(900\right)\\ =\$34200\\ P\left(600,1800\right)=12\left(600\right)+18\left(1800\right)\\ =\$39600\end{array}$

Thus, the number of snack size and family size bags that must be processed in order to maximize profitis 600 and 1800 respectively.