maxivite healthovite total
iron 40 10 >= 2200
vitamin b1 12 18 >= 2100
vitamin b2 8 18 >= 1800
let x = the number of maxivite pills.
let y = the number of healthovite pills.
each maxivite pill costs .06
each healthivite pill costs .08
you want to minimize cost, therefore your objective function is .06x + .08y
your constraints are:
40x + 10y >= 2200 (iron constraints)
12x + 18y >= 2100 (vitamin b1 constraints)
8x + 18y >= 1800 (vitamin b2 constraints)
x >= 0
y >= 0
those last 2 constraints are implied because the number of pills can't be negative.
in this problem, they may be redundant, but in other problems, they can be necessary to ensure the answer is not a negative quantity.
the desmos.com calculator at https://www.desmos.com/calculator makes graphing these inequalities easy.
you just graph the opposite of the inequalities and find the area of the graph that is NOT shaded.
that will be your feasible region.
you evaluate your objective function at the corner points of this feasible region to find the corner point that gives you the lowest cost.
the inequalities that will be graphed are the opposite of the constraint inequalities as shown below.
40x + 10y <= 2200 (opposite of iron constraints)
12x + 18y <= 2100 (opposite of vitamin b1 constraints)
8x + 18y <= 1800 (opposite of vitamin b2 constraints)
x <= 0
y <= 0
with this calculator, you also don't need to solve for y.
you can just enter the inequalities as is.
here's the output from this calculator.
you evaluate your objective function at each of these corner points.
your objective function is .06x + .08y
at (0,220), your cost = 17.6
at (31,96), your cost = 9.54
at (75,66.67), your cost = 9.83
at (225,0), your cost = 13.5
your minimum cost is at (31,96)
you then evaluate your constraint functions at this corner point.
you will get 2200 mg of iron and 2100 mg of vitamin b1 and 1976 mg of vitamin b2.
since you need >= 1800 mg of vitamin b2, you will be getting more than the minimum specified.
you will get exactly the minimum specified for iron and vitamin b1.
as an example, at (31,96), you will get 8 * 31 = 248 mg of vitamin b1 from your maxivite pills + 18 * 96 mg = 1728 of vitamin b2 from your healthovite pills for a total of 1976 mg of vitamin b2.
the problem does not specify the time period in which these pills need to be taken nor does it specify the order in which these pills need to be taken, therefore no assumption is made concerning that.
this is strictly a what is the lowest cost combination of pills that can be taken that meed the constraints specified.