To solve this problem, we need to formulate a system of linear equations based on the given information and then solve for the number of units of product A and product B that should be manufactured to maximize profit.

Let:

• $x_1$ = number of units of Product A
• $x_2$ = number of units of Product B

The factory constraints based on the table are as follows:

• Machine I: $1x_1 + 2x_2 \leq 72$
• Machine II: $2x_1 + 2x_2 \leq 98$
• Machine III: $4x_1 + 2x_2 \leq 180$

Objective:

Maximize profit $P$, where: $P = 45x_1 + 50x_2$

Step 1: Set up the constraints

$1x_1 + 2x_2 \leq 72$ $2x_1 + 2x_2 \leq 98$ $4x_1 + 2x_2 \leq 180$

Step 2: Solve the system

We can solve this linear programming problem using the graphical method, or by applying a method like the simplex algorithm. Let me compute the optimal solution for you using a simplex method approach.

I'll proceed with the necessary calculations to determine the values of $x_1$ and $x_2$.To maximize profit, the factory should produce approximately 26 units of Product A and 23 units of Product B.

The maximum profit would be $2320.

Would you like more details on any specific step or have further questions?

Here are 5 related questions to deepen your understanding:

1. How does the simplex method work in solving linear programming problems?
2. What would happen if the profit per unit for Product B increased?
3. How can sensitivity analysis be applied to this problem?
4. Could we use another method, like graphical analysis, to solve this?
5. How does changing machine availability impact the production quantities?

Tip: When using linear programming for optimization, always double-check that the constraints reflect real-world limitations, like availability of resources.