Q. 24.1( 55 Votes )
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)
Rate this question :
A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines, and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at ` 7 profit and that of B at a profit of ` 4. Find the production level per day for maximum profit graphically.Mathematics - Board Papers
A retired person wants to invest an amount of ₹ 50,000. His broker recommends investing in two types of bonds ‘A’ and ‘B’ yielding 10% and 9% return respectively on the invested amount. He decides to invest at least ₹ 20,000 in bond ‘A’ and at least ₹ 10,000 in bond ‘B’. He also wants to invest at least as much in bond ‘A’ as in bond ‘B’. Solve this linear programming problem graphically to maximize his returns.Mathematics - Board Papers
A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs80 on each piece of type A, and Rs120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?Mathematics - Board Papers