Question

In a factory, there are two machines $A$ and $B$ producing toys. They respectively produce $60$ and $80$ units in one hour. $A$ can run a maximum of $10$ hours and $B$ a maximum of $7$ hours a day. The cost of their running per hour respectively amounts to $2,000$ and $2,500$ rupees. The total duration of working these machines cannot exceed $12$ hours a day. If the total cost cannot exceed Rs. $25,000$ per day and the total daily production is at least $800$ units, then formulate the problem mathematically.

Answer 1

Let $x$ and $y$ be the daily production units of machines $A$ and $B$ respectively.

The total daily production is atleast $800$.

$\Rightarrow x+y\ge 800$

$A$ can produce $60$ units per hour for a maximum of $10$ hours.

$\Rightarrow x\le 60\times 10$

$\Rightarrow x\le 600$

$B$ can produce $80$ units per hour for a maximum of $7$ hours.

$\Rightarrow y\le 80\times 7$

$\Rightarrow y\le 560$

The number of hours machine $A$ runs is $\frac{x}{60}$ and machine $B$ runs is $\frac{y}{80}$.

Hence, total cost per day $=$ Rs. $(2000\frac{x}{60}+2500\frac{y}{80})=$ Rs. $(\frac{100x}{3}+\frac{125y}{4})\le$ Rs. $25000$

Hence, the mathematical formulation of the given problem will be:

$\frac{100x}{3}+\frac{125y}{4}\le 25000$

$x+y\ge 800$

$x\le 600$

$y\le 560$