## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Laura', 'hours worked by Paul', 'hours worked by Bill', 'hours worked by Mary'] which can be represented symbolically as 'x0', 'x1', 'x2', 'x3' respectively.

## Step 2: Create a dictionary mapping symbolic variables to their natural language descriptions
The symbolic variables and their corresponding natural language descriptions are:
- 'x0': 'hours worked by Laura'
- 'x1': 'hours worked by Paul'
- 'x2': 'hours worked by Bill'
- 'x3': 'hours worked by Mary'

## Step 3: Define the objective function in symbolic notation
The objective function to maximize is $9x_0 + 6x_1 + 5x_2 + 3x_3$.

## Step 4: List all the constraints in symbolic notation
Constraints:
1. $10x_0 \geq 10x_0$ (always true, but explicitly given)
2. $6x_0 \geq 6x_0$ (always true, but explicitly given)
3. $9x_0 \geq 9x_0$ (always true, but explicitly given)
4. $7x_0 \geq 7x_0$ (always true, but explicitly given)
5. $12x_1 \geq 12x_1$ (always true, but explicitly given)
6. $8x_1 \geq 8x_1$ (always true, but explicitly given)
7. $8x_1 \geq 8x_1$ (always true, but explicitly given)
8. $14x_1 \geq 14x_1$ (always true, but explicitly given)
9. $6x_2 \geq 6x_2$ (always true, but explicitly given)
10. $7x_2 \geq 7x_2$ (always true, but explicitly given)
11. $12x_2 \geq 12x_2$ (always true, but explicitly given)
12. $9x_2 \geq 9x_2$ (always true, but explicitly given)
13. $14x_3 \geq 14x_3$ (always true, but explicitly given)
14. $11x_3 \geq 11x_3$ (always true, but explicitly given)
15. $3x_3 \geq 3x_3$ (always true, but explicitly given)
16. $1x_3 \geq 1x_3$ (always true, but explicitly given)
17. $10x_0 + 14x_3 \geq 46$
18. $10x_0 + 6x_2 \geq 49$
19. $12x_1 + 6x_2 + 14x_3 \geq 30$
20. $8x_1 + 3x_3 \geq 47$
21. $9x_0 + 12x_2 + 3x_3 \geq 49$
22. $7x_0 + 9x_2 \geq 66$
23. $14x_1 + 9x_2 \geq 27$
24. $9x_2 + 1x_3 \geq 47$
25. $10x_0 + 6x_2 \leq 142$
26. $12x_1 + 14x_3 \leq 232$
27. $12x_1 + 6x_2 \leq 153$
28. $10x_0 + 6x_2 + 14x_3 \leq 172$
29. $10x_0 + 12x_1 + 6x_2 \leq 138$
30. $10x_0 + 12x_1 + 6x_2 + 14x_3 \leq 138$
31. $6x_0 + 7x_2 \leq 129$
32. $6x_0 + 8x_1 \leq 179$
33. $6x_0 + 11x_3 \leq 195$
34. $8x_1 + 11x_3 \leq 240$
35. $8x_1 + 7x_2 \leq 164$
36. $8x_1 + 7x_2 + 11x_3 \leq 174$
37. $6x_0 + 8x_1 + 7x_2 + 11x_3 \leq 174$
38. $8x_1 + 12x_2 \leq 177$
39. $8x_1 + 3x_3 \leq 185$
40. $9x_0 + 8x_1 + 12x_2 + 3x_3 \leq 185$
41. $7x_0 + 9x_2 \leq 272$
42. $9x_2 + 1x_3 \leq 146$
43. $14x_1 + 1x_3 \leq 222$
44. $7x_0 + 14x_1 + 9x_2 + 1x_3 \leq 222$

## 5: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Laura
x1 = m.addVar(name="x1", lb=0)  # hours worked by Paul
x2 = m.addVar(name="x2", lb=0)  # hours worked by Bill
x3 = m.addVar(name="x3", lb=0)  # hours worked by Mary

# Define the objective function
m.setObjective(9 * x0 + 6 * x1 + 5 * x2 + 3 * x3, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(10 * x0 + 14 * x3 >= 46)
m.addConstr(10 * x0 + 6 * x2 >= 49)
m.addConstr(12 * x1 + 6 * x2 + 14 * x3 >= 30)
m.addConstr(8 * x1 + 3 * x3 >= 47)
m.addConstr(9 * x0 + 12 * x2 + 3 * x3 >= 49)
m.addConstr(7 * x0 + 9 * x2 >= 66)
m.addConstr(14 * x1 + 9 * x2 >= 27)
m.addConstr(9 * x2 + 1 * x3 >= 47)
m.addConstr(10 * x0 + 6 * x2 <= 142)
m.addConstr(12 * x1 + 14 * x3 <= 232)
m.addConstr(12 * x1 + 6 * x2 <= 153)
m.addConstr(10 * x0 + 6 * x2 + 14 * x3 <= 172)
m.addConstr(10 * x0 + 12 * x1 + 6 * x2 <= 138)
m.addConstr(10 * x0 + 12 * x1 + 6 * x2 + 14 * x3 <= 138)
m.addConstr(6 * x0 + 7 * x2 <= 129)
m.addConstr(6 * x0 + 8 * x1 <= 179)
m.addConstr(6 * x0 + 11 * x3 <= 195)
m.addConstr(8 * x1 + 11 * x3 <= 240)
m.addConstr(8 * x1 + 7 * x2 <= 164)
m.addConstr(8 * x1 + 7 * x2 + 11 * x3 <= 174)
m.addConstr(6 * x0 + 8 * x1 + 7 * x2 + 11 * x3 <= 174)
m.addConstr(8 * x1 + 12 * x2 <= 177)
m.addConstr(8 * x1 + 3 * x3 <= 185)
m.addConstr(9 * x0 + 8 * x1 + 12 * x2 + 3 * x3 <= 185)
m.addConstr(7 * x0 + 9 * x2 <= 272)
m.addConstr(9 * x2 + 1 * x3 <= 146)
m.addConstr(14 * x1 + 1 * x3 <= 222)
m.addConstr(7 * x0 + 14 * x1 + 9 * x2 + 1 * x3 <= 222)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Laura: ", x0.varValue)
    print("Hours worked by Paul: ", x1.varValue)
    print("Hours worked by Bill: ", x2.varValue)
    print("Hours worked by Mary: ", x3.varValue)
else:
    print("The model is infeasible")
```

```json
{
    'sym_variables': [
        ['x0', 'hours worked by Laura'], 
        ['x1', 'hours worked by Paul'], 
        ['x2', 'hours worked by Bill'], 
        ['x3', 'hours worked by Mary']
    ], 
    'objective_function': '9*x0 + 6*x1 + 5*x2 + 3*x3', 
    'constraints': [
        '10*x0 + 14*x3 >= 46',
        '10*x0 + 6*x2 >= 49',
        '12*x1 + 6*x2 + 14*x3 >= 30',
        '8*x1 + 3*x3 >= 47',
        '9*x0 + 12*x2 + 3*x3 >= 49',
        '7*x0 + 9*x2 >= 66',
        '14*x1 + 9*x2 >= 27',
        '9*x2 + 1*x3 >= 47',
        '10*x0 + 6*x2 <= 142',
        '12*x1 + 14*x3 <= 232',
        '12*x1 + 6*x2 <= 153',
        '10*x0 + 6*x2 + 14*x3 <= 172',
        '10*x0 + 12*x1 + 6*x2 <= 138',
        '10*x0 + 12*x1 + 6*x2 + 14*x3 <= 138',
        '6*x0 + 7*x2 <= 129',
        '6*x0 + 8*x1 <= 179',
        '6*x0 + 11*x3 <= 195',
        '8*x1 + 11*x3 <= 240',
        '8*x1 + 7*x2 <= 164',
        '8*x1 + 7*x2 + 11*x3 <= 174',
        '6*x0 + 8*x1 + 7*x2 + 11*x3 <= 174',
        '8*x1 + 12*x2 <= 177',
        '8*x1 + 3*x3 <= 185',
        '9*x0 + 8*x1 + 12*x2 + 3*x3 <= 185',
        '7*x0 + 9*x2 <= 272',
        '9*x2 + 1*x3 <= 146',
        '14*x1 + 1*x3 <= 222',
        '7*x0 + 14*x1 + 9*x2 + 1*x3 <= 222'
    ]
}
```