let

x = the number of type A printers to order

y = the number of type B printers to order.

You expect to sell at least 100 laser printers this month

x + y >= 100

you need to make at least $3150 profit on them

45x + 30y >= 3450

Minimize Cost = 165x + 110y

So the problem is to Minimize Cost = 165x + 110y where

x + y >= 100

45x + 30y >= 3450

x > 0

y > 0

The last two come from the fact that x and y can't be negative.

They limit the graph to the 1st quadrant, i.e., above the x-axis and to the right of the y-axis.

Now we form the boundary equations by replacing the inequality symbol in each to an equal sign:

x + y = 100 [intercepts (0,100) and (100,0]

45x + 30y = 3450 [intercepts (0,115) and (76 2/3, 0)

x = 0

y = 0

Draw the graphs and find all the corner points of this region.

They are (0,115), (30,70), and (100,0)

The minimum cost will be the smallest of the values found for Cost when Cost = 165x + 110y is evaluated at each of these corner points:

For (0,115), Cost = 165x + 110y = 165(0) + 110(115) = $12650

For (30,70), Cost = 165x + 110y = 165(30) + 110(70) = $12650

For (100,0), Cost = 165x + 110y = 165(100) + 110(0) = $16500

The minimum of these is $12650. That's the answer to (b). Now, since two adjacent corner points happen to yield this same minimum cost, then any point on the line that connects them where both x and y are non-negative integers will yield that same minimum cost, namely $12650. So there are 15 possible answers:

(2,112) which means to order 2 A's and 112 B's

(4,109) which means to order 4 A's and 109 B's

...

(26,76) which means to order 26 A's and 76 B's

(28,73) which means to order 28 A's and 73 B's

(30,70) which means to order 30 A's and 70 B's