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प्रश्न
A company manufactures 2 types of goods P and Q that requires copper and brass. Each unit of type P requires 2 grams of brass and 1 gram of copper while one unit of type Q requires 1 gram of brass and 2 grams of copper. The company has only 90 grams of brass and 80 grams of copper. Each unit of types P and Q brings profit of ₹ 400 and ₹ 500 respectively. Find the number of units of each type the company should produce to maximize its profit
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उत्तर
Let the company manufacture 'x' units of type P and 'y' units of type Q.
Since number of goods cannot be negative, x ≥ 0, y ≥ 0
Each unit of types P and Q brings profit of ₹ 400 and ₹ 500 respectively.
∴ Total profit = Z = 400x + 500y
We construct a table with the constraints of Brass and Copper as follows:
| Material/Type | P | Q | Maximum Availability |
| Brass | 2 | 1 | 90 |
| Copper | 1 | 2 | 80 |
From the table, the constraints are
2x + y ≤ 90
x + 2y ≤ 80
∴ Given problem can be formulated as follows:
Maximize Z = 400x + 500y
Subject to 2x + y ≤ 90
x + 2y ≤ 80, x ≥ 0, y ≥ 0
To draw the feasible region, construct table as follows:
| Inequality | 2x + y ≤ 90 | x + 2y ≤ 80 |
| Corresponding equation (of line) | 2x + y = 90 | x + 2y = 80 |
| Intersection of line with X-axis | (45, 0) | (80, 0) |
| Intersection of line with Y-axis | (0, 90) | (0, 40) |
| Region | Origin side | Origin side |
Shaded portion OABC is the feasible region, whose vertices are O(0, 0), A(45, 0), B and C(0, 40).
B is the point of intersection of the lines 2x + y = 90 and x + 2y = 80. Solving the above equations, we get
x = `100/3`, y `70/3`
∴ B ≡ `(100/3, 70/3)`
Here, the objective function is
Z = 400x + 500y
∴ Z at O(0,0) = 400(0) + 500(0) = 0
Z at A(45, 0) = 400(45) + 500(0) = 18000
Z at B`(100/3, 70/3) = 400(100/3) + 500(70/3)`
= `40000/3 + 35000/3`
= `75000/3`
= 25000
Z at C(0, 40) = 400(0) + 500(40)
= 20000
∴ Z has maximum value 25000 at x = `100/3` and y `70/3`
∴ The company should produce `100/3` units of type P and `70/3` units of type Q to get maximum profit of ₹ 25000.
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