Please answer 1-9

Ana is a university student with 12 hours available daily, and she wants to distribute her time between study (x) and leisure (y) activities. She wants to maximize her total enjoyment while fulfilling her university obligations. Ana considers leisure to be twice (2) as enjoyable as studying, but she also believes that she should spend at least twice (2) as much time studying as she does on leisure activities. Additionally, she cannot spend more than five (5) hours a day on leisure activities to ensure that she has enough time for her university obligations.

The following questions need to be answered:

1. Formulate the optimization model for this problem: define the decision variables, objective function, and constraints.

2. Solve the problem graphically: draw the feasible region, and evaluate its corner points to find the optimal solution

3. How many hours should Ana spend on leisure activities (x?) and how many on studying (y?) to maximize her enjoyment while fulfilling her university obligations?

4. What is Ana's maximum enjoyment (objective function) given her time constraints and preferences?

5. Classify the constraints in terms of active and non-active.

6. How much additional enjoyment would Ana get if she had one more hour per day available, so 13 instead of 12 (dual price)?

7. Would Ana experience a greater level of enjoyment if she allows herself to have a maximum of six hours per day to leisure activities instead of five, while still fulfilling her university obligations within the 12 hours available (dual price)?

8. What is the allowable increase of the coefficient accompanying leisure in the objective function (i.e., the coefficient equal to 2 in the objective function)?

9. Explain the meaning of the number referenced in question 8 using the problem context?