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  • In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligrams per tablet) are given as below: Tablets Iron Calcium Vitamin X 6 3 2 Y 2 3 4 The person n

Q: In order to supplement daily diet, a person wishes to take some X and some wishes Y tablets. The contents of iron, calcium and vitamins in X and Y (in milligrams per tablet) are given as below:

\begin{array}{|l|l|l|l|} \hline \text { Tablets } & \text { Iron } & \text { Calcium } & \text { Vitamin } \\ \hline \mathrm{X} & 6 & 3 & 2 \\ \hline \mathrm{Y} & 2 & 3 & 4 \\ \hline \end{array}

The person needs at least 18 milligrams of iron, 21 milligrams of calcium and 16 milligrams of vitamins. The price of each tablet of X and Y is Rs 2 and Re 1 respectively. How many tablets of each should the person take in order to satisfy the above requirement at the minimum cost?

Answers (1)

Let us say that the number of tablet X be x and the number of tablet Y be y.

Iron content in X and Y tablets is 6 mg and 2 mg respectively.

Total iron content from x and y tablets = 6x + 2y

Minimum of 18 mg of iron is required. So, we have

\\ 6x + 2y \geq $ 18\\ \\ 3x + y $ \geq 9\\

Similarly, calcium content in X and Y tablets is 3 mg each respectively.

So, total calcium content from x and y tablets = 3x + 3y

Minimum of 21 mg of calcium is required. So, we have

\\ 6x + 2y \geq $ 21\\ \\ $ \Rightarrow $ x + y $ \geq 7\\

Also, vitamin content in X and Y tablet is 2 mg and 4 mg respectively.

So, total vitamin content from x and y tablets = 2x + 4y

Minimum of 16 mg of vitamin is required. So, we have

\\ 2x + 4y \geq $ 16\\ \\ $ \Rightarrow $ x + 2y $ \geq 8\\ \\

Also, as number of tablets should be non-negative so, we have,

x, y ≥ 0

Cost of each tablet of X and Y is Rs 2 and Re 1 respectively.

Let total cost = Z

So, Z = 2x + y

Finally, we have,

Constraints,

\\ 3x + y \geq $ 9\\ \\ x + y $ \geq $ 7\\ \\ x + 2y $ \geq $ 8\\ \\ x, y $ \geq 0\\ \\ Z = 2x + y\\

We need to minimize Z, subject to the given constraints.

Now let us convert the given inequalities into equation.

We obtain the following equation

\\ 3x + y \geq $ 9\\ \\ $ \Rightarrow $ 3x + y = 9\\ \\ x + y $ \geq $ 7\\ \\ $ \Rightarrow $ x + y = 7\\ \\ x + 2y $ \geq $ 8\\ \\ $ \Rightarrow $ x + 2y = 8\\ \\ x $ \geq $ 0\\ \\ $ \Rightarrow $ x=0\\ \\ y $ \geq $ 0\\ \\ $ \Rightarrow y=0\\ \\

The region that is representing 3x + y ≥ 9 is the line 3x + y = 9 meets the coordinate axes (3,0) and (0,9) respectively. Once the points are joined, the lines are obtained 3x + y = 9. It does not satisfy the inequation and so the region that contains the origin then represents the solution set of the inequation 3x + y ≥ 9.

The region that is representing x + y ≥ 7 is the line x + y = 7 meets the coordinate axes (7,0) and (0,7) respectively. Once the points are joined, the lines are obtained v It does not satisfy the inequation and so the region that contains the origin then represents the solution set of the inequation x + y ≥ 7.

The region representing x + 2y ≥ 8 is the line x + 2y = 8 meets the coordinate axes (8,0) and (0,4) respectively. We will join these points to obtain the line x + 2y = 8. It is clear that (0,0) does not satisfy the inequation x + 2y ≥ 8. So, the region not containing the origin represents the solution set of the inequation x + 2y ≥ 8.

The regions that represent x≥0 and y≥0 is first quadrant, since every point in the first quadrant satisfies these inequations.

Given below is the graph:

The region towards the right of ABCD is the feasible region. It is unbounded in this case.

Value of Z at corner points A,B,C and D :

\begin{array}{|l|l|} \hline \text { Corner Point } & \text { Value of } z=2 x+y \\ \hline A(0,9) & Z=0+9=9 \\ \hline B(1,6) & Z=1(2)+6=8 \rightarrow \text { min } \\ \hline C(6,1) & Z=6(2)+1=13 \\ \hline D(8,0) & Z=8(2)+0=16 \\ \hline \end{array}

Now, we check if 2 x+y<8 to check if resulting open half has any point common with feasible region.

The region represented by 2 x+y<8:

The line 2x + y=8meets the coordinate axes (4,0) and (0,8) respectively. We will join these points to obtain the line 2x + y=8. It is clear that (0,0) satisfies the inequation 2 x+y<8. So, the region not containing the origin represents the solution set of the inequation 2 x+y<8.

Clearly, 2x + y=8 intersects feasible region only at B.
So, 2 x+y<8 does not have any point inside feasible region.
So, value of Z  is minimum at B(1,6), the minimum value is 8 .
So, number of tablets that should be taken of type X and Y is 1,6 respectively.

 

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