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A manufacturer makes two types of toys $\mathbf{A}$ and $\mathbf{B}$. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type $A$ is $₹ 7.50$ and that on each toy of type $B$ is ₹ 5 , show that 15 toys of type $A$ and 30 of type $B$ should be manufactured in a day to get maximum profit.
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type $A$ is $₹ 7.50$ and that on each toy of type $B$ is ₹ 5 , show that 15 toys of type $A$ and 30 of type $B$ should be manufactured in a day to get maximum profit.
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Let $\mathrm{x}$ toys of type A and $\mathrm{y}$ toys of type $\mathrm{B}$ are manufactured. $\therefore \max ^{\mathrm{m}}(\mathrm{Z})=7.5 \mathrm{x}+5 \mathrm{y}$
Subject to constraints are $12 x+6 y \leq 360$,
$18 x \leq 360,6 x+9 y \leq 360$
i.e., $2 \mathrm{x}+\mathrm{y} \leq 60, \mathrm{x} \leq 20,2 \mathrm{x}+3 \mathrm{y} \leq 120$ and also $\mathrm{x}$, $y \geq 0$
Now $\mathrm{Z}=7 \cdot 5 \mathrm{x}+5 \mathrm{y}$.
$$
\begin{aligned}
&\text { At } \mathrm{P}(15,30) \quad \mathrm{Z}=7 \cdot 5 \times 15+5 \times 30 \\
&\quad=112 \cdot 5+150=262 \cdot 5 \rightarrow \text { max }^{\mathrm{m}}
\end{aligned}
$$
$\Rightarrow \mathrm{Z}$ is maximum when $\mathrm{x}=40, \mathrm{y}=160$
$\Rightarrow 40$ tickets of executive class and 160 tickets of economy class should be sold to get the maximum profit of $₹ 13600$.
Subject to constraints are $12 x+6 y \leq 360$,
$18 x \leq 360,6 x+9 y \leq 360$
i.e., $2 \mathrm{x}+\mathrm{y} \leq 60, \mathrm{x} \leq 20,2 \mathrm{x}+3 \mathrm{y} \leq 120$ and also $\mathrm{x}$, $y \geq 0$
Now $\mathrm{Z}=7 \cdot 5 \mathrm{x}+5 \mathrm{y}$.
$$
\begin{aligned}
&\text { At } \mathrm{P}(15,30) \quad \mathrm{Z}=7 \cdot 5 \times 15+5 \times 30 \\
&\quad=112 \cdot 5+150=262 \cdot 5 \rightarrow \text { max }^{\mathrm{m}}
\end{aligned}
$$
$\Rightarrow \mathrm{Z}$ is maximum when $\mathrm{x}=40, \mathrm{y}=160$
$\Rightarrow 40$ tickets of executive class and 160 tickets of economy class should be sold to get the maximum profit of $₹ 13600$.
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