Question

An aeroplane can carry a maximum of two passengers. A profit of Rs. 1000 is made on each executive clas ticket and a profit of Rs. 600 is made on each economy class ticket. The airline reserves at least 20 seats for executive class. However at least 4 times as many passengers prefer to travel by economy clas than by the executive class. Determine how many tickets of each type must be sold in order to maximise the profit for the airline. What is the maximum profit?

Answer 1

Let `x` passengers are of executive class and `y` passengers are of economy class.

Maximise `Z=1000x+600y`……………1
and constraints `x+yle200`………………2
`xge20`…………….3
`y-4xge0impliesyge4x`…………..4
`xge0,yge0`............5
First draw the graph of the line `x+y=200`.

Put `(0,0)` in the inequation `x+yle200`,
`0+0le200implies0le200`. (True)
Thus, half plane contains the origin.
Now, draw the graph of line `y=4x`

Put `(10,0)` in the inequation `yge4x`,
`0ge4xx10implies0ge40` (False)
Thus, half plane will be opposite to `(10,0)`.
Now draw the graph of line `x=20`
Put `(0,0)` in the inequation `xge0,0ge20` (False)
Thus, the half plane does not contain origin.
Since `x,yge0`, So the feasible region will be in first quadrant.
From the equations the points of intersection are `A(20,80),B(40,160)` and `C(20,180)`.
`:.` Feasible region is ABCA
The vertices of the feasible region are `A(20,80),B(40,160)` and `C(20,180)` at which we find the value of `Z`.

Thus, the maximum value of `Z` is 136000 at point `B(40,160)`. Therefore, maximum cost is Rs. 136000 for which 40 tickets of executive class and 160 tickets of economy class are required.