x = number of hours the A plant is in production
y = hours the H plant is in production
z = hours the M plant is in production
Regular Ice Cream production in gallons =
40 x gal + 20 y gal + 40 z gal ≥ 250 gal
Premium Ice cream production in gallons =
20 x gal + 40 y gal + 40 z gal ≥ 150 gal
Cost = $50 x + $59 y + $100 z
You have three variables and only two equations, one for regular and the other for premium, the cost function is NOT a equation you can use to solve the system because you have no data for it. You must first solve the system in order to have the data to evaluate, and minimize, the cost function. The set of all answers to the system is the domain of the cost function.
Fortunately, we can eliminate the complicating feature of this problem.
Notice that if plants A and H both work for an hour, they will produce 60 gal of Regular and 60 gal of Premium Ice Cream for a cost of $109. For plant M to meet that order it would have to operate for 1.5 hours for a cost of $150.
Thus, the optimal solution will not use plant M at all because it cannot complete with A and H working together.
So now our two equations, in two variables, are:
40 x + 20 y ≥ 250
20 x + 40 y ≥ 150
Let us solve
20y = -40 x + 250
y = -2x + 12.5
40y = -20x + 150
y = -1/2 x + 3.75
these lines will intersect at
-2x + 12.5 = -1/2x + 3.75
8.75 = 3/2 x
5.83 = x
y = -2(5.83) + 12.5 = 0.83
Since you are supposed to round both answers to the nearest tenth, it is important that the rounding does not cost you the goal of producing the correct minimum amount of Ice Cream of either variety.
40(5.8) + 20(0.8) ≥ 250
232 + 16 ≥ 250 is wrong
40(5.8) + 20(0.9) ≥ 250
232 + 18 ≥ 250 works
40(5.9) + 20(0.8) ≥ 250
236 + 16 ≥ 250 works as well, and costs less (the extra 0.1 hours cost $5, not $5.90)
20(5.8) + 40(0.8) ≥ 150
116 + 32 ≥ 150 does not work
20(5.9) + 40(0.8) ≥ 150
118 + 32 ≥ 150 does work (although this is the low cost option for Regular Ice Cream, it was still necessary to prove its sufficiency for Premium Ice Cream.)
The other extreme values will be
x ≥ 7.5, y = 0 (7.5 hours require to meet the Premium quota is only plant A is running.)
y ≥ 12.5, x = 0 (12.5 hours required to meet the Regular Ice Cream quota if only plant H is running.)
Our cost equation is now
C= $50x + $59y
Our boundary values are (7.5, 0) (0, 12.5) and (5.9, 0.8)
C(7.5, 0) = $50 (7.5) + $59 (0) = $375
C(0, 12.5) = $50 (0) + $59 (12.5) = $737.50
C(5.9, 0.8) = $50 (5.9) + $59 (0.8) = $295 + $47.20 = $342.20
So 5.9 hours at plant A and 0.8 at plant H is the lowest-cost production schedule.
