Question

One kind of cake requires 200g 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 5 kg of flour and 1 kg of fat assuming that there is no shortage of the other ingredients used in making the cakes?

Answer 1

Assume that the cake of one kind be x and another kind be y. As the quantities are always positive, so,

x≥0 y≥0

Tabulate the given data as,

	Flour(g)	Fat(g)
Cakes of first kind, x	200	25
Cakes of second kind, y	100	50
Availability	5000	1000

The required constraints are,

200x+100y≤5000 2x+y≤50

And,

25x+50y≤1000 x+2y≤40

The objective function which needs to maximize is,

Z=x+y

The line 2x+y≤50 gives the intersection point as,

x	0	25
y	50	0

Also, when x=0,y=0 for the line 2x+y≤50, then,

0+0≤50 0≤50

This is true, so the graph have the shaded region towards the origin.

The line x+2y≤40 gives the intersection point as,

x	0	40
y	20	0

Also, when x=0,y=0 for the line x+2y≤40, then,

0+0≤40 0≤40

This is true, so the graph have the shaded region towards the origin.

By the substitution method, the intersection points of the lines 2x+y≤50 and x+2y≤40 is ( 30,10 ).

Plot the points of all the constraint lines,

It can be observed that the corner points are O( 0,0 ),A( 25,0 ),B( 20,10 ),C( 0,20 ).

Substitute these points in the given objective function to find the minimum value of Z.

Corner Points	Z=x+y
O( 0,0 )	0
A( 25,0 )	25
B( 20,10 )	30 (maximum)
C( 0,20 )	20

The maximum value of Z is 30 at the point ( 20,10 ).

Therefore, the maximum numbers of cakes that can be made are 30.