Question

An aeroplane can carry a maximum of $200$ passengers. A profit of $Rs.1000$ is made on each executive class 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 class 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's assume that executive class ticket sold be X and economy class ticket sold be Y.

Since, aeroplane can carry maximum 200 passengers.

$\therefore X+Y\le 200...\left(1\right)$

Since, atleast 20 tickets is reserves for executive class.

$\therefore X\ge 20...\left(2\right)$

Since the number of tickets for economy class should be at least 4 times the executive class.

$\therefore Y\ge 4X\Rightarrow Y-4X\ge 0...\left(3\right)$

Also, the number of tickets can't be negative.

So, $X\ge 0andY\ge 0...\left(4\right)$

Profit on an executive class ticket is $1000Rs$ and profit on an economy class ticket is $600Rs$

So, Total profit $\left(Z\right)=1000X+600Y$

We have to maximize the total profit. After plotting all the constraints given by equation (1), (2), (3) and (4), we get the feasible region as shown in the image.

Corner points	Value of $Z=1000X+600Y$
A (20, 180)	128000
B (40, 160)	136000 (Maximum)
C (20,80)	68000

Hence, maximum profit will be $136000Rs$, when number of executive class ticket sold be $40$ and number of economy class ticket sold be $160$