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# 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

2. 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.

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Let there be x cakes of first kind and y cakes of the second kind.Thus, $x\geq 0,y\geq 0$.

The given information can be represented in the table as :

 Flour(g) fat(g) Cake of kind  x 200 25 Cake of kind y 100 50 Availability 5000 1000

Therefore,

$200x+100y\leq 5000$

$\Rightarrow \, \, \, \, 2x+y\leq 50$

. $\, \, 25x+50y\leq 10000$

$\Rightarrow \, \, x+2y\leq 400$

The total number of cakes, Z.   Z=X+Y

Subject to constraint,

$\Rightarrow \, \, \, \, 2x+y\leq 50$

$\Rightarrow \, \, x+2y\leq 400$

$x\geq 0,y\geq 0$

The feasible  region determined by constraints is as follows:

The corner points of the feasible region are $A(25,0),B(20,10),C(0,20),D(0,0)$

The value of Z at corner points is as shown :

 corner points Z=X+Y $A(25,0)$ 25 $B(20,10)$ 30 maximum $C(0,20)$ $D(0,0)$ 20               0 minimum

The maximum cake can be made 30 (20 of the first kind and 10 of the second kind).

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