Q2 of 31 Page 519

One kind of cake requires 200g of flour and 25g of fat, and another kind of cake requires 100g of flour and 50g of fat. Find the maximum number of cakes which can be made from 5kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes.

let there be x cakes of first kind and y cakes of second kind.

∴ x≥ 0 and y ≥ 0.


The information given in the question can be complied in given form



200x + 100y ≤ 5000 ⇒ 2x+ y ≤ 50


& 25x + 50y ≤ 1000 ⇒ x +2y ≤ 40


Let Z be the total number of cakes that can be made


⇒ Z =X=y


Mathematical formulation of the given problem is


Maximize Z =x+y


Subject to constraint 2x+ y ≤ 50 and x +2y ≤ 40 where x, y ≥ 0


The graphical representation shows the feasible region determined by the system of constraints.



The corner points A(25, 0) , B( 20,10), 0(0,0) and C(0,20)


The values of z at these corner points are as follows



Thus, the maximum number of cakes that can be made are 30 (20 of one kind and 10 of other kind)


More from this chapter

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10

Maximize Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0.

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Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs 80/kg. Food P contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food Q contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture.

 3

A factory makes tennis rackets and cricket bats. A tennis racket takes 1.5 hours of machine time and 3 hours of craftman’s time in its making while a cricket bat takes 3 hour of machine time and 1 hour of craftman’s time. In a day, the factory has the availability of not more than 42 hours of machine time and 24 hours of craftsman’s time.

(i) What number of rackets and bats must be made if the factory is to work at full capacity?


(ii) If the profit on a racket and on a bat is Rs 20 and Rs 10 respectively, find the maximum profit of the factory when it works at full capacity.

 4

A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs17.50 per package on nuts and Rs 7.00 per package on bolts. How many packages of each should be produced each day so as to maximise his profit, if he operates his machines for at the most 12 hours a day?