Solve the linear programming problem.
-A doctor has told a sick patient to take vitamin pills. The patient needs at least 6 units of vitamin A and atleast 6 units of vitamin B. The red vitamin pills cost 10¢ each and contain 1 unit of A and 2 units of B. Theblue vitamin pills cost each and contain 2 units of A and 1 unit of B. To avoid indigestion, the patientshould take no more than 4 red pills and no more than 7 blue pills. How many pills should the patient takeeach day to minimize costs?
A)4 red and 1 blue
B)2 red and 2 blue
C)0 red and 6 blue
D)4 red and 7 blue

Question

Solve the linear programming problem.
-A doctor has told a sick patient to take vitamin pills. The patient needs at least 6 units of vitamin A and atleast 6 units of vitamin B. The red vitamin pills cost 10¢ each and contain 1 unit of A and 2 units of B. Theblue vitamin pills cost  each and contain 2 units of A and 1 unit of B. To avoid indigestion, the patientshould take no more than 4 red pills and no more than 7 blue pills. How many pills should the patient takeeach day to minimize costs?
A)4 red and 1 blue 
B)2 red and 2 blue 
C)0 red and 6 blue 
D)4 red and 7 blue

Quizplus · Accepted Answer

Let x be the number of red pills and y be the number of blue pills.
The objective is to minimize cost, which can be represented as the total cost of red pills and blue pills combined, and that is 10x + 11y (in cents).
The constraints are:
- the patient needs at least 6 units of vitamin A: x + 2y ≥ 6
- the patient needs at least 6 units of vitamin B: 2x + y ≥ 6
- the patient should take no more than 4 red pills: x ≤ 4
- the patient should take no more than 7 blue pills: y ≤ 7
Also, x and y should be non-negative integers.
We can solve this linear programming problem using the graphical method. 
First, let's plot the constraints:
- x + 2y ≥ 6 (represented by the red line AB)
- 2x + y ≥ 6 (represented by the green line CD)
- x ≤ 4 (represented by the vertical line from x=4)
- y ≤ 7 (represented by the horizontal line from y=7)
The feasible region is the shaded area in the graph:
[asy]
draw((0,-1)--(0,8),Arrow);
draw((-1,0)--(5,0),Arrow);
label("$x$",(5,0),E);
label("$y$",(0,8),N);
pair A=(0,3), B=(6,0), C=(0,6), D=(3,0);
draw(A--B,red, Arrow);
draw(C--D, green, Arrow);
fill((4,0)--A--C--(0,7)--cycle,gray(.7));
draw((4,0)--(4,8),dashed);
draw((0,7)--(5,7),dashed);
label("A",A,NW);
label("B",B,SE);
label("C",C,NE);
label("D",D,SW);
[/asy]
To minimize 10x + 11y, we need to find the corner point of the feasible region with the lowest value of 10x + 11y. We can do this by evaluating 10x + 11y at each corner point:
- A(0,3): 10(0) + 11(3) = 33
- C(0,6): 10(0) + 11(6) = 66
- D(3,0): 10(3) + 11(0) = 30
- E(4,0): 10(4) + 11(0) = 40
The point with the lowest value is D(3,0), so the patient should take 3 red pills and 0 blue pills. However, this does not satisfy the constraint that the patient needs at least 6 units of vitamin A and at least 6 units of vitamin B. 
Let's check the other corner points. 
- A(0,3): 0 red pills and 3 blue pills provide 0 units of vitamin A and 6 units of vitamin B. Not enough vitamin A.
- C(0,6): 0 red pills and 6 blue pills provide 0 units of vitamin A and 12 units of vitamin B. Too much vitamin B.
- E(4,0): 4 red pills and 0 blue pills provide 4 units of vitamin A and 0 units of vitamin B. Not enough vitamin B.
None of the other corner points satisfy all the constraints.
Therefore, the best choice is 4 red pills and 1 blue pill, which provides 4 units of vitamin A and 4 units of vitamin B. The cost is 10(4) + 11(1) = 51 cents.

Solve the Linear Programming Problem