This is a linear programming problem that maximizes profit subject to resource constraints. Let’s break it down:

Definitions:

• Black and White TVs: Let $x$ be the number of black and white TVs produced.

  • Cost per set: $2700$
  • Profit per set: $525$

• Colored TVs: Let $y$ be the number of colored TVs produced.

  • Cost per set: $3600$
  • Profit per set: $675$

Constraints:

1. Total Production: $x + y = 200$ (exactly 200 sets are made per week).
2. Total Cost Constraint: $2700x + 3600y \leq 648,000$ (maximum weekly cost).
3. Non-Negativity: $x \geq 0, y \geq 0$.

Objective Function:

Maximize profit $P$: $P = 525x + 675y$

Step 1: Express $y$ in terms of $x$ from the total production constraint:

$y = 200 - x$

Step 2: Substitute $y = 200 - x$ into the total cost constraint:

$2700x + 3600(200 - x) \leq 648,000$ Simplify: $2700x + 720,000 - 3600x \leq 648,000$ $-900x + 720,000 \leq 648,000$ $-900x \leq -72,000$ $x \leq 80$

Thus, $x \leq 80$.

Step 3: Substitute $y = 200 - x$ into the profit function:

$P = 525x + 675(200 - x)$ Simplify: $P = 525x + 135,000 - 675x$ $P = -150x + 135,000$

Step 4: Maximize profit within the feasible range of $x$:

• From the constraints, $x \leq 80$ and $x + y = 200$. Therefore:
  • If $x = 80$, then $y = 200 - 80 = 120$.
  • Profit $P = -150(80) + 135,000 = 123,000$.

Final Answer:

The maximum profit is Rs. 1,23,000, which corresponds to Option (1).

Would you like me to explain any step further or solve a similar problem?

Here are 5 questions you could explore next:

1. How do we solve when there are more than two variables?
2. How can graphical methods be used to solve linear programming problems?
3. What if the total production constraint is not an equality but an inequality?
4. How does increasing the cost constraint affect the solution?
5. Can we solve this problem using software tools like Excel or Python?

Tip: Always ensure constraints are carefully interpreted, especially equality or inequality signs.