Sintok Quarry Co. has two quarries with different production capabilities for producing the same type of gravel. After mining and crushing, the gravel is separated into three sizes - large, medium and small. The company has contracted to provide the highway contractor with 24 tons of large gravel, 16 tons of medium gravel and 48 tons of small size gravel each week.It costs Sintok Quarry Co. RM 10,000 per day to operate Quarry A and RM5,000 per day to operate Quarry B. In one day, Quarry A produces 6 tons of large size gravel, 2 tons of medium size gravel and 4 tons of small size gravel.Quarry B produces 2, 2 and 12 tons per day of each gravel size, respectively. Sintok Quarry would like to know how many DAYS per week to operate each quarry that can save them the overall cost.

1. Formulate a linear programming model for this problem.
2. Solve this model using graphical method.
3. Prove your answer using mathematical analysis. Explain your answer.
4. If the production is at its optimum (answer in b and c) how much, if any, excess production for any gravel size?