A factory produces two articles, A and B, each requiring processing on two machines, X and Y.

- Article A requires 2 hours on machine X and 4 hours on machine Y.
- Article B requires 4 hours on machine X and 2 hours on machine Y.

If \( x \) is the number of article A and \( y \) is the number of article B produced daily, write down two inequalities in \( x \) and \( y \), given that neither machine X nor Y can operate more than 24 hours a day.

Additionally, if all articles produced are sold, and each article A yields a profit of Rs. 60 while each article B yields a profit of Rs. 100, determine how many of each article should be produced daily to maximize profit.

To write down two inequalities in x and y, we can consider the constraints given in the problem. In this case, we have A requiring 2 hours of X and 4 hours of Y, and B requiring 4 hours of X and 2 hours of Y. To find the number of each article that should be produced daily for maximum profit, we can set up a profit equation considering the profit each article yields.

Explanation:

To write down two inequalities in x and y, we need to consider the constraints given in the problem. We know that A requires 2 hours of X and 4 hours of Y, while B requires 4 hours of X and 2 hours of Y. To ensure that neither X nor Y works more than 24 hours a day, we can write the following inequalities:

2x + 4y ≤ 24
4x + 2y ≤ 24

To find the number of each article that should be produced daily for maximum profit, we need to consider the profit each article yields. A yields a profit of Rs. 60, while B yields a profit of Rs. 100. Let P represent the total profit. We can express this as:

P = 60x + 100y

Now, we can solve the system of inequalities along with the profit equation to find the values of x and y that maximize the profit P.