Let the required number of trees of Type A and B be Rs x and Rs y respectively.
Number of trees cannot be negative.
x, y ≥ 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,
Type A costs Rs 20 per tree and Type B costs Rs 25 per tree. Therefore, x trees of type A and y trees of type B cost Rs 20x and Rs 25y respectively. A man has a sum of Rs 1400 to purchase young trees.
Thus the mathematical formulation of the given LPP is
Max Z = 40x - 20x + 60y - 25y = 20x + 35y
Subject to,
Region 4x + 5y ≤ 280: line 4x + 5y ≤ 280 meets axes at A1(70, 0), B1(0, 56) respectively.
The region containing origin represents 4x + 5y ≤ 280 as (0, 0) satisfies 4x + 5y ≤ 280.
Region x + 2y ≤ 100: line x + 2y = 100 meets axes at A2(100, 0), B2(0, 50) respectively.
Region containing origin represents x + 2y ≤ 100 as (0, 0) satisfies x + 2y ≤ 100
Region x, y ≥ 0: it represents the first quadrant.
The corner points are A1(70, 0), P(20, 40), B2(0, 50)
The values of Z at these corner points are as follows:
The maximum value of Z is 1800 which is attained at P(20,40).
Thus the maximum profit is Rs 1800 obtained when Rs 20 were involved in Type A and Rs 40 were involved in Type II.