Question

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?

Answer 1

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 formulat​ion 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.