Question

A man owns a field of area 1000 sq.m. He wants to plant fruit trees in it. He has a sum of 1400 to purchase young trees. He has the choice of two type 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?

Answer 1

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.