Question

One kind of cake requires 200 g of flour and 25 g of fat and another kind of cake requires 100 g of flour and 50 g 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

Let x be number of cakes of one kind and y be the number of cakes of other kind. Construct the following table :

$\begin{array}{cccc}Kind& Numberofcakes& Flourrequired\left(ing\right)& Fatrequired\\ & & \left(ing\right)& \left(ing\right)\\ I& x& 200x& 25x\\ II& y& 100y& 50y\\ Total& x+y& 200x+100y& 25x+50y\\ Requirement& & 5000& 1000\end{array}$

Our problem is to maximize Z = x + y ......(i)

Subject to constraints are $200x+100y\le 5000\iff 2x+y\le 50$ .............(ii)

$25x+50y\le 1000\iff x+2y\le 40$ .........(iii)

and x, y $\ge$ 0 ...........(iv)

Firstly, draw the graph of the lines 2x + y = 50

$\begin{array}{ccc}x& 0& 25\\ y& 50& 0\end{array}$

Putting (0, 0) in the inequality 2x + y $\le$ 50, we have

$2\times 0+0\le 50\Rightarrow 0\le 50$ (which is true)

So, the half plane is towards the origin.

Secondly, draw the graph of the line x + 2y = 40

$\begin{array}{ccc}x& 0& 40\\ y& 20& 0\end{array}$

Putting (0, 0) in the inequality x + 2y $\le$ 40, we have

$0+2\times 0\le 40$

$\Rightarrow 0\le 40$ (which is true)

So, the half plane is towards the origin.

Since, x, y $\ge$ 0

So, the feasible region lies in the first quadrant.

On solving equations $2x+y=50$ and x + 2y = 40, we get B(20, 10)

$\therefore$ Feasible region is OABCO.

The corner points of the feasible region are 0(0, 0), A(25, 0), B(20, 10) and C(0, 20). The values of Z at these points are as follows:

$\begin{array}{cc}& \\ Cornerpoint& Z=x+y\\ 0(0,0)& 0\\ A(25,0)& 25\\ B(20,10)& 30\to Maximum\\ C(0,20)& 20\end{array}$

Thus, the maximum number of cakes that can be made is 30 i.e., 20 of one kind and 10 of the other kind .