To solve this linear programming problem graphically, we need to determine the optimal number of products $A$ and $B$ to maximize profit, under the constraints provided by the raw material usage and the condition that at least as many products $B$ are made as $A$.

Step 1: Define the Variables

Let:

• $x_1$ = the number of product $A$ units produced,
• $x_2$ = the number of product $B$ units produced.

Step 2: Objective Function

The profit function is given by: $Z = 30x_1 + 40x_2$ where:

• 30 is the profit from one unit of $A$,
• 40 is the profit from one unit of $B$.

Step 3: Constraints

The constraints are based on the consumption of the three types of raw materials:

1. Material I: $12x_1 + 4x_2 \leq 300$
2. Material II: $4x_1 + 4x_2 \leq 120$
3. Material III: $3x_1 + 12x_2 \leq 252$ Additionally, there is a condition that at least as many $B$ units should be produced as $A$: $x_2 \geq x_1$ Finally, we have the non-negativity constraint: $x_1 \geq 0, \quad x_2 \geq 0$

Step 4: Solve Graphically

To solve the problem graphically, we plot each of the constraints on a graph with $x_1$ on the x-axis and $x_2$ on the y-axis. The area where all constraints overlap will be the feasible region.

Steps for plotting:

1. Convert each inequality into an equation and plot the line:
   • For Material I: $12x_1 + 4x_2 = 300$,
   • For Material II: $4x_1 + 4x_2 = 120$,
   • For Material III: $3x_1 + 12x_2 = 252$.
2. Shade the region that satisfies all inequalities.
3. Add the line $x_2 = x_1$ to ensure the condition that at least as many $B$ units are produced as $A$.
4. Identify the vertices of the feasible region.
5. Calculate the objective function $Z$ at each vertex to find the maximum.

Would you like me to plot the graph for this problem, or would you prefer further guidance on how to proceed with this solution?

Here are 5 questions that expand on this:

1. How do you determine which regions on the graph satisfy each inequality?
2. How do you calculate the coordinates of the intersection points of the constraint lines?
3. How can we verify the feasibility of each vertex in the solution space?
4. What if the constraint $x_2 \geq x_1$ was reversed? How would this affect the solution?
5. Can the simplex method offer a more efficient solution for larger systems of inequalities?

Tip: When solving linear programming problems graphically, always check the vertices of the feasible region, as the optimal solution will always lie at one of these points.