A manufacturer makes two types of handmade fancy paper bags. Type A and Type B. Two designers a cutter and a finisher need to work on both kinds of bags. A type A bag requires 2 hours of the cutters time and 3 hours of the finishers time. A type B bag requires 3 hours of the cutters time and 1 hour of the finishers time. Each month the cutter is available for 108 hours and the finisher is available 78 hours. The manufacturer gets a profit of $12 for each bag of type A and $9 for each bag of type B. Identify the number of bags for each type to be manufactured to obtain the maximum profit.

The graph shows the constraints and the boundaries of the feasible region. The maximum profit will be had with the manufacture of 18 Type A bags and 24 Type B bags.

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The inequality describing the constraint on cutter hours is ...

2a +3b ≤ 108

The inequality describing the constraint on finisher hours is ...

3a +1b ≤ 78

The boundary lines of the solution regions of these inequalities intersect at ...

(a, b) = (18, 24)

The profit function is such that it doesn't pay to make all of one type or the other bags. The most profit is had for the mix ...

18 Type A bags; 24 Type B bags.

On the graph, the line representing the profit function will be as far as possible from the origin at the point of maximum profit.