A vanilla cake requires $200 g$ of flour and $25 g$ of fat, and a strawberry cake requires $100 g$ of flour and $50 g$ of fat. From $5 kg$ of flour and $1 kg$ of fat, the maximum number of cakes that can be made is

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Solution

Let $x$ number of vanilla cakes and $y$ number of strawberry cakes be made.
Then objective function is
Maximize $Z = x + y$
subject to constraints : maximum $5 kg$ flour is available.
$\Rightarrow \frac{200}{1000} x + \frac{100}{1000} y \leq 5$
$\Rightarrow 2 x + y \leq 50 \dots (i)$
and maximum $1 kg$ of fat is available.
$\Rightarrow \frac{25}{1000} x + \frac{50}{1000} \leq 1$
$\Rightarrow x + 2 y \leq 40 \dots (i i)$
and $x, y \geq 0 \dots (i i i)$
Now, plotting the graph of given constraints to get feasible region :

Now, tabulating the value of $Z = x + y$ at corner points :

Corner point : $(x, y)$ Value : $Z = x + y$

$(0, 0)$ $0$

$(0, 20)$ $20$

$(25, 0)$ $25$

$(20, 10)$ $30$ (maximum)

So, $Z_{max} = 30$

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