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Table 1 shows the Labor (in hours per unit) and amount of Clay (Lb. per unit) for producing
a bowl and a mug by Company X in a single day.
If Company X plans to maximise the Profit, and the limitations of clay available is
100Lb/day, the number of hours available is 40hours/day,
a) Identify the variables involved (2 Marks

To maximize the profit for Company X, we need to identify the variables involved in the production process.

### Variables Involved

1. **Decision Variables**:
   - **$ x $**: The number of bowls produced per day.
   - **$ y $**: The number of mugs produced per day.

2. **Resource Constraints**:
   - **Labor Hours Constraint**: The total labor hours available per day is limited to 40 hours.
   - **Clay Constraint**: The total clay available per day is limited to 100 pounds.

3. **Profit per Unit**:
   - **Profit from Bowls**: $40 per bowl.
   - **Profit from Mugs**: $50 per mug.

The objective is to determine the values of $ x $ (bowls produced) and $ y $ (mugs produced) that maximize the total profit while adhering to the constraints on labor and clay.

Would you like more details on how to set up the equations for maximizing profit?

Here are 5 questions that further explore this problem:

1. How would you formulate the objective function for maximizing profit based on the identified variables?
2. What would be the equations representing the labor and clay constraints?
3. How can you solve the system of inequalities to find the optimal production quantities for bowls and mugs?
4. What is the significance of a feasible region in this context?
5. How would changes in the availability of clay or labor hours affect the solution?

**Tip:** Understanding how to translate word problems into mathematical equations is key to solving optimization problems effectively.

Explore how to maximize profit for Company X by optimizing production of bowls and mugs within labor and clay constraints. Learn about setting up the objective function and solving resource constraints using linear programming techniques. Suitable for advanced high school students and anyone interested in mathematical optimization problems.

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