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Home » A man owns a field area . He wants to plant fruit trees in it. He has a sum of to purchase young trees. He has the choice of two types of trees. Type A requires of ground per trees and costs per tree, and type B requires of ground per tree and costs per tree. When full grown, a type A tree produces an average of of fruit which can be sold at a profit per and type tree produces an average of of fruit which can be sold at a profit of per . How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?
A man owns a field area 1000 \mathrm{~m}^{2}. He wants to plant fruit trees in it. He has a sum of \pm 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 \mathrm{~m}^{2} of ground per trees and costs \pm 20 per tree, and type B requires 20 \mathrm{~m}^{2} of ground per tree and costs \pm 25 per tree. When full grown, a type A tree produces an average of 20 \mathrm{~kg} of fruit which can be sold at a profit \pm 2 per \mathrm{kg} and type -\mathrm{B} tree produces an average of 40 \mathrm{~kg} of fruit which can be sold at a profit of \pm 1.50 per \mathrm{kg}. How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?
CBSE | Class 12 | Exercise 33B | Linear Programming | Maths | RS Aggarwal
A man owns a field area 1000 \mathrm{~m}^{2}. He wants to plant fruit trees in it. He has a sum of \pm 1400 to purchase young trees. He has the choice of two types of trees. Type A requires 10 \mathrm{~m}^{2} of ground per trees and costs \pm 20 per tree, and type B requires 20 \mathrm{~m}^{2} of ground per tree and costs \pm 25 per tree. When full grown, a type A tree produces an average of 20 \mathrm{~kg} of fruit which can be sold at a profit \pm 2 per \mathrm{kg} and type -\mathrm{B} tree produces an average of 40 \mathrm{~kg} of fruit which can be sold at a profit of \pm 1.50 per \mathrm{kg}. How many of each type should be planted to achieve maximum profit when tree are full grown? What is the maximum profit?

Let x and y be number of A and B trees.
\therefore According to the question,
20 x+25 y \leq 1400,10 x+20 y \leq 1000, x \geq 0, y \geq 0
Maximize Z=40 x+60 y
The feasible region determined by 20 x+25 y \leq 1400,10 x+20 y \leq 1000, x \geq 0, y \geq 0 is given by

The corner points of feasible region are A(0,0), B(0,50), C(20,40), D(70,0).The value of Z at corner points are

    \[\begin{tabular}{|l|l|l|} \hline Corner Point & $Z=40 x+60 y$ & \\ \hline $\mathrm{A}(0,0)$ & 0 \\ \hline $\mathrm{B}(0,50)$ & 3000 & \\ \hline $\mathrm{C}(20,40)$ & 3200 & \\ \hline $\mathrm{D}(70,0)$ & 2800 \\ \hline \end{tabular}\]

The maximum value of Z is 3200 at point (20,40).
Hence, the man should plant 20 A trees and 40 B trees to make maximum profit of Rs. 3200 .

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