A patient takes vitamin pills. Each day he must have at least 540IU of vitamin​ A, 28mg of vitamin Upper B1​, and 80mg of vitamin C.....

Question

A patient takes vitamin pills. Each day he must have at least 540IU of vitamin​ A, 28mg of vitamin Upper B1​, and 80mg of vitamin C. He can choose between pill​ 1, which contains

120IU of vitamin​ A, 4mg of vitamin

Upper B1​, and 10mg of vitamin​ C, and pill​ 2, which contains

60IU of vitamin​ A, 4mg of vitamin

Upper B 1​, and 20mg of vitamin C. Pill 1 costs 15cents¢​, and pill 2 costs 45cents¢.

Complete parts a and b below.

he should buy ____of pill 1

he should buy ____ of pill 2

the minimum cost is ___

Answer 1

Shorthand used in solution:

A represents vitamin A

B represents vitamin upper B1

C represents vitamin C

Let:

x = qty of pilll1 pills (we are asked to solve for x)

y = qty of pilll2 pills (we are asked to solve for y)

cost_x = cost of one x (one pill1) which the problem says is 0.15

cost_y = cost of one y (one pill2) which the problem says is 0.45

need_a = how many IU we need of A which the problem says is 540

need_B = how many IU we need of B which the problem says is 28

need_c = how many IU we need of C which the problem says is 80

a_x = IU of A provided by one x (one pill1) which the problem says is 120

b_x = IU of B provided by one x (one pill1) which the problem says is 4

c_x = IU of C provided by one x (one pill1) which the problem says is 10

a_y = IU of A provided by one x (one pill1) which the problem says is 60

b_y = IU of B provided by one x (one pill1) which the problem says is 4

c_y = IU of C provided by one x (one pill1) which the problem says is 20

tip: you may want to copy the entire solution to a word document and find/replace cost_x, cost_y, need_a, need_b, need_c, a_x, b_x, c_x, a_y, b_y and c_y with full word descriptions if that makes this more readable for you.

Remember that x must be greater than or equal to zero and y must be greater than or equal to zero (no negative pill counts!)

Build a chart with these columns: x, y, IUs of A, IUs of B, IUs of C, sufficient: yes or no, total cost

Build rows of data as follows (if you use excel this will be much easier to do):

First row of data:

X=0

y = 1

IUs of A = (the x for this row * a_x) + (the y for this row 0* a_y)

IUs of B = (the x for this row * b_x) + (the y for this row 0* b_y)

IUs of C = (the x for this row * c_x) + (the y for this row 0* c_y)

Sufficient is “YES” if IUs of A for this row >=need_a AND IUs of B for this row >=need_b AND IUs of C for this row>=need_c else Sufficient is “NO”

Total cost = the x for this row *cost_x+ the y for this row *cost_y

Note: if sufficient = YES then for the next row add 1 to X and set y = 0, if sufficient = NO then for the next row keep x the same and add 1 to y

Repeat the calculations above for: IUs, Sufficient yes or no, and total cost. So again…

IUs of A = (the x for this row * a_x) + (the y for this row 0* a_y)

IUs of B = (the x for this row * b_x) + (the y for this row 0* b_y)

IUs of C = (the x for this row * c_x) + (the y for this row 0* c_y)

Sufficient is “YES” if IUs of A for this row >=need_a AND IUs of B for this row >=need_b AND IUs of C for this row>=need_c else Sufficient is “NO”

Total cost = the x for this row *cost_x+ the y for this row *cost_y

Keep going until you notice that x keeps incrementing and Y remains zero, you can stop at that point.

Take a moment to review your table and understand what you have calculated, and what it is showing you.

At this point, if you are doing this with pen and paper, scan your table for the lowest total cost and Sufficient = “YES”. That is your answer. It turns out to be x = 8, y =0, total cost = 1.2

If you are using excel, I recommend you copy your worksheet to another worksheet, then Control-A to select ALL cells, copy-> paste special-> values so you get rid of your formulas, sort by “sufficient” then “total cost”, and go to the row where Sufficient = YES and your total cost is the least total cost, it turns out to be x = 8, y =0, total cost = 1.2.

Second way to approach this follows. It requires knowing how to graph systems of inequalities. I will, for sake of brevity, assume you are able to graph systems of inequalities, you may want to research youtube for videos explaining how to graph systems of inequalities.

Total_cost = 0.15x + .45y

540 >= 120 x + 60 y

28 >= 4 x + 4 y

80 >= 10 x + 20 y

therefore

540 - 120 x >= 60 y

28 - 4 x >= 4 y

80 - 10 x >= 20 y

therefore

9 - 2 x >= y

7 - x >= y

4 - 0.5 x >= y

therefore

y >= 9 - 2x

y >= 7 - x

y >= 4 - x/2

go to www.symbolab.com/solver/system-of-inequalities-calculator

put y >= 9 - 2x, y >= 7 - x, y >= 4 - x/2 in the box next to the red GO button and press GO

scroll down to graph solution and your darkest shaded area is your solution area.

You need to be in top right quadrant which is the quadrant where both x are y non-negative (no negative pill counts!)

You will notice that you have four “boundary” points for the darkest shaded area, those are your four possible answers (you may need to “zoom in” to read the graph)

x Y

0 9

8 0

6 1

2 5

Use the equations above to confirm that your IU values are sufficient, and to determine the lowest total cost

x Y IU A IU B IU C total_cost

0 9 540 36 180 4.05

8 0 960 32 80 1.2

6 1 780 28 80 1.35

2 5 540 28 120 2.55

Again, your answer is x = 8, y =0, total cost = 1.2.

Ideally, take advantage of your study/homework time to understand a math problem in more than one ways, as this will help you feel more familiar and comfortable with the type of question. If you get a similar question during the exam, start with the approach that is quickest for you until you get an answer. If you don’t have time to do the problem a different way, quickly check your work, but, ideally, if you have time to do the problem a different way, do the problem a different way and check that your answers match.