One kind of cake requires $300 g m$ of flour and $15 g m$ of fat and another kind of cake requires $150 g m$ of flour and $30 g m$ of fat. Find the maximum number of cake that can be made from $7.5 k g$ of flour and $600 g m$ of fat . Form a linear programming problem and solve it graphically.

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Solution

Let $x$ and $y$ be the number of cake of first and second kind respectively.
Case I:- Given that first kind of cake requires $300 g$ of flour while the second kind of cake required $150 g$ of flour.
Maximum quantity available $= 7.5 k g = 7500 g$
Therefore,
$300 x + 150 y \leq 7500$
$2 x + y \leq 50; x \geq 0, y \geq 0$
$x$ $0$ $25$
$y$ $50$ $0$
Case II:- Given that first kind of cake requires $15 g$ of fat while second kind of cake required $30 g$ of fat.
Maximum quantity available $= 600 g$
Therefore,
$15 x + 30 y \leq 600$
$x + 2 y \leq 40; x \geq 0, y \geq 0$
$x$ $0$ $40$
$y$ $20$ $0$
Total no. of cakes $= x + y$
$M a x Z = x + y$
Corner points Value of $Z$
$(0, 20)$ $20$
$(20, 10)$ $30$
$(25, 0)$ $25$
Thus value of $Z$ is maximum when $x = 20$ and $y = 30$ , i.e.,
No. of cakes will be maximum when the no. of cakes of first and second kind are $20$ and $10$ respectively.
Hence the maximum no. of cakes that can be made are $30$ .

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Corner points	Value of $Z$
$(0, 20)$	$20$
$(20, 10)$	$30$
$(25, 0)$	$25$

One kind of cake requires 300gm of flour and 15gm of fat and another kind of cake requires 150gm of flour and 30gm of fat. Find the maximum number of cake that can be made from 7.5kg of flour and 600gm of fat . Form a linear programming problem and solve it graphically.

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