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प्रश्न
A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of Rs 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 sq.m of ground per tree and costs Rs 20 per tree and type B requires 20 sq.m of ground per tree and costs Rs 25 per tree. When fully grown, type A produces an average of 20 kg of fruit which can be sold at a profit of Rs 2.00 per kg and type B produces an average of 40 kg of fruit which can be sold at a profit of Rs. 1.50 per kg. How many of each type should be planted to achieve maximum profit when the trees are fully grown? What is the maximum profit?
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उत्तर
Let the man planted x trees of type A and y trees of type B.
Number of trees cannot be negative.
Therefore, \[x, y \geq 0\] To plant tree of type A requires 10 sq.m and type B requires 20 sq.m of ground per tree. And, it is given that a man owns a field of area 1000 sq.m.Therefore,
\[10x + 20y \leq 1000\]
Type A costs Rs 20 per tree and type B costs Rs 25 per tree. Therefore, x trees of type Aand y trees of type B costs Rs 20x and Rs 25y respectively. A man has a sum of Rs 1400 to purchase young trees.
\[20x + 25y \leq 1400\]
Thus, the mathematical formulation of the given linear programmimg problem is
Max Z = 40x − 20x + 60y − 25y = 20x + 35y
subject to
\[10x + 20y \leq 1000\]
\[20x + 25y \leq 1400\]
The feasible region determined by the system of constraints is 
The corner points are A(0, 50), B(20, 40), C(70, 0)
The values of Z at these corner points are as follows
| Corner point | Z = 20x + 35y |
| A | 1750 |
| B | 1800 |
| C | 1400 |
The maximum value of Z is 1800 which is attained at B(20, 40)
Thus, the maximum profit is Rs 1800 obtained when Rs 20 were invested on type A and Rs 40 were invested on type B.
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