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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?
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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