Q22 of 118 Page 31

An aeroplane can carry a maximum of 200 passengers. A profit of ₹ 400 is made on each first class ticket and a profit of ₹ 600 is made on each economy class ticket. The airline reserves at least 20 seats of first class. However, at least 4 times as many passengers prefer to travel by economy class to the first class. Determine how many each type of tickets must be sold in order to maximize the profit for the airline. What is the maximum profit.

Let required number of first class and economy class tickets be x and y respectively.


Each ticket of first class and economy class make profit of Rs 400 and Rs 600 respectively.


So, x ticket of first class and y tickets of economy class make profit of Rs 400x and Rs 600y respectively.


Let total profit be Z = 400x + 600y


Given, aeroplane can carry a minimum of 200 passengers, so,


x + y 200


Given airline reserves at least 20 seats for first class, so,


x 20


Also, at least 4 times as many passengers prefer to travel by economy class to the first class, so


y 4x


Hence the mathematical formulation of the LPP is


Max Z = 400x + 600y


Subject to constraints


x + y 200


y 4x


x 20


x,y 0 [ since seats in both the classes can not be zero]


Region represented by x + y 200: the line x + y = 200 meets the axes at A(200,0), B(0,200). Region containing origin represents x + y 200 as (0,0) satisfies x + y 200.


Region represented by x 20: line x = 20 passes through (20,0) and is parallel to y axis. The region to the right of the line x = 20 will satisfy the inequation x 20


Region represented by y 4x: line y = 4x passes through (0,0). The region above the line y = 4x will satisfy the inequation y 4x


Region x,y 0: it represents the first quadrant.


22.jpg


The corner points are C(20,80), D(40,160), E(20,180).


The values of Z at these corner points are as follows:



The maximum value of Z is attained at E(20, 180).


Thus, the maximum profit is Rs 116000 obtained when 20 first class tickets and 180 economy class tickets are sold.


More from this chapter

All 118 →
20

A manufacturer produces two type of steel trunks. He has two machines A and B. For completing, the first types of the trunk requires 3 hours on machine A and 3 hours on machine B, whereas the second type of the trunk requires 3 hours on machine A and 2 hours on machine B . Machines A and B can work at most for 18 hours and 15 hours per day respectively. He earns a profit of Rs 30 and Rs 25 per trunk of the first type and the second type respectively. How many trunks of each type must
he make each day to make maximum profit?

 21

A manufacturer of patent medicines is preparing a production plan on medicines, A and B. There are sufficient raw materials available to make 20000 bottles of A and 40000 bottles of B, but there are only 45000 bottles into which either of the medicines can be put. Further, it takes 3 hours to prepare enough material to fill 1000 bottles of A, it takes 1 hours to prepare enough material to fill 1000 bottles of B and there are 66 hours available for this operation. The profit is ₹ 8 per bottle for A and ₹ 7 per bottle for B. How should the manufacturer schedule his production in order to maximize his profit?

 23

A gardener has a supply of fertilizer of type I which consists of 10% nitrogen and 6% phosphoric acid and type II fertilizer which consists of 5% nitrogen and 10% phosphoric acid. After testing the soil conditions, he finds that he needs at least 14 kg of nitrogen and 14 kg of phosphoric acid for his crop. If the type I fertilizer costs 60 paise per kg and type II fertilizer costs 40 paise per kg, determine how many kilograms of each fertilizer should be used, so that nutrient requirements are met at a minimum cost. What is the minimum cost?

 24

Anil wants to invest at most ₹ 12000 in Saving Certificates and National Saving Bonds. According to rules, he has to invest at least ₹ 2000 in Saving Certificates and at least 4000 in National Saving Bonds. If the rate of interest on saving certificate is 8% per annum and the rate of interest on National Saving Bonds is 10% per annum, how much money should he invest to earn

maximum yearly income? Find also his maximum yearly income.