﻿ Examples

### Question

A firm exports two types of machine P and Q. Type P occupies 2m3 of space and type Q, 4 m3. Type P weighs 9kgs and type Q 6kgs. The available shipping space is 1,600m3 and the total weight of the machines cannot exceed 3,600 kgs. The profit on type P is €100 and the profit on type Q is €80. How many of each machine must be exported to bring a maximum profit and what is the profit.

Let x represent the number of type P exported and y represent the number of Q exported.

therefore 2x + 4y<= 1,600 which on dividing across by 2 gives x+ 2y<=800
9x + 6y<=3,600 which on dividing across by 3 gives 3x + 2y <= 1200

The other two inequalities will be x>=0 and y>=0 as you cannot have a negative no of machines.

The four inequalities are

1. x>=0
2. y>=0
3. x + 2y<=800 substituting x=0 and y=0 we get the points (0,400) and (800,0)
4. 3x + 2y<=1200 substituting x=0 and y=0 we get the points (0,600) and (400,0)

When 3 and 4 are solved simultaneously we get the point (200,300). Now we draw the lines and use the value (0,0) to indicate the region represented by each inequality.Click here to see the two steps again.

The following is the required diagram.

The profit on P is €100 and the profit on Q is €80 therefore the profit is 100x + 80y.

Following Step 4 of finding the maximum and minimun values we construct a table for simplicity.

 Vertex 100x 80y 100x +80y (0,0) 0 0 0 (0,400) 0 32,000 32,000 (200,300) 20,000 24,000 44,000 (400,0) 40,000 0 40,000

From the table we can see that the company should export 200 machines of type P and 300 machines of type Q to get a Maximum value of €44,000.