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प्रश्न
In order to supplement daily diet, a person wishes to take X and Y tablets. The contents (in milligrams per tablet) of iron, calcium and vitamins in X and Y are given as below :
| Tablets | Iron | Calcium | Vitamin |
| x | 6 | 3 | 2 |
| y | 2 | 3 | 4 |
The person needs to supplement 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 Rs 1 respectively. How many tablets of each type should the person take in order to satisfy the above requirement at the minimum cost? Make an LPP and solve graphically.
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उत्तर
Let the person take x tablets of type X and y tablets of type Y.
According to the given condition, the person requires at least 18 milligrams of iron.
\[\therefore 6x + 2y \geq 18 \Rightarrow 3x + y \geq 9 . . . . . \left( 1 \right)\]
Also, the person needs atleast 21 milligrams of calcium.
\[\therefore 3x + 3y \geq 21 \Rightarrow x + y \geq 7 . . . . . \left( 2 \right)\]
The person also needs atleast 16 mg of vitamins.
\[\therefore 2x + 4y \geq 16 \Rightarrow x + 2y \geq 8 . . . . . \left( 3 \right)\]
Hence, the given linear programming problem is
Minimise Z = 2x + y
subject to the constraints
3x + y ≥ 9
x + y ≥ 7
x + 2y ≥ 8
and x, y ≥ 0
The region represented by the system of inequations given in constraints is shown as the shaded region.
This shaded region repressents the feasible region of the given linear programming problem. The corner points of the feasible region are A(0, 9), B(1, 6), C(6, 1) and D(8, 0).
The values of the objective function at these points are given in the following table.
| Corner points | Cost (Z = 2x + y) |
| A(0, 9) | Z = 2 × 0 + 9 = 9 |
| B(1, 6) | Z = 2 × 1 + 6 = 8 |
| C(6, 1) | Z = 2 × 6 + 1 = 13 |
| D(8, 0) | Z = 2 × 8 + 0 = 16 |
So, Z is minimum at x = 1 and y = 6.
Hence, the person should take 1 tablet of type X and 6 tablet of type Y in order to meets the minimum requirements at the minimum cost.
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