## Step 1: Define the variables and their corresponding symbolic notation
Let's define the variables and their corresponding symbolic notation:
- 'hours worked by Ringo' = $x_1$
- 'hours worked by Mary' = $x_2$
- 'hours worked by Laura' = $x_3$
- 'hours worked by Jean' = $x_4$
- 'hours worked by Hank' = $x_5$
- 'hours worked by John' = $x_6$

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: $6.8x_1 + 4.4x_2 + 4.92x_3 + 1.59x_4 + 7.47x_5 + 2.36x_6$

## Step 3: List the constraints in symbolic notation
Constraints:
1. $10x_1 \leq 283$
2. $5x_2 \leq 283$
3. $2x_3 \leq 283$
4. $2x_4 \leq 283$
5. $10x_5 \leq 283$
6. $6x_6 \leq 283$
7. $10x_1 + 2x_3 \geq 47$
8. $2x_4 + 10x_5 \geq 47$
9. $2x_4 + 6x_6 \geq 47$
10. $10x_5 + 6x_6 \geq 38$
11. $5x_2 + 10x_5 \geq 27$
12. $2x_3 + 2x_4 \geq 37$
13. $10x_1 + 2x_3 + 2x_4 \geq 37$
14. $5x_2 + 2x_3 + 6x_6 \geq 37$
15. $10x_1 + 5x_2 + 2x_3 \geq 37$
16. $10x_1 + 5x_2 + 6x_6 \geq 37$
17. $10x_1 + 2x_3 + 2x_4 \geq 38$
18. $5x_2 + 2x_3 + 6x_6 \geq 38$
19. $10x_1 + 5x_2 + 2x_3 \geq 38$
20. $10x_1 + 5x_2 + 6x_6 \geq 38$
21. $10x_1 + 2x_3 + 2x_4 \geq 37$
22. $5x_2 + 2x_3 + 6x_6 \geq 37$
23. $10x_1 + 5x_2 + 2x_3 \geq 37$
24. $10x_1 + 5x_2 + 6x_6 \geq 37$
25. $10x_1 + 2x_3 + 2x_4 \geq 29$
26. $5x_2 + 2x_3 + 6x_6 \geq 29$
27. $10x_1 + 5x_2 + 2x_3 \geq 29$
28. $10x_1 + 5x_2 + 6x_6 \geq 29$
29. $x_2 - 9x_5 + 5x_6 \geq 0$
30. $5x_2 + 10x_5 \leq 144$
31. $5x_2 + 2x_4 \leq 163$
32. $10x_1 + 2x_4 \leq 176$
33. $2x_3 + 6x_6 \leq 225$
34. $5x_2 + 10x_5 + 6x_6 \leq 59$
35. $10x_1 + 5x_2 + 2x_4 \leq 105$
36. $10x_1 + 2x_3 + 6x_6 \leq 133$
37. $2x_3 + 2x_4 + 10x_5 \leq 56$
38. $10x_1 + 5x_2 + 10x_5 \leq 223$
39. $5x_2 + 2x_4 + 10x_5 \leq 192$
40. $5x_2 + 2x_3 + 2x_4 \leq 107$
41. $2x_3 + 2x_4 + 6x_6 \leq 188$
42. $10x_1 + 2x_3 + 2x_4 \leq 108$
43. $10x_1 + 5x_2 + 2x_3 + 2x_4 + 10x_5 + 6x_6 \leq 108$

## Step 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Ringo'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Laura'),
        ('x4', 'hours worked by Jean'),
        ('x5', 'hours worked by Hank'),
        ('x6', 'hours worked by John')
    ],
    'objective_function': '6.8x1 + 4.4x2 + 4.92x3 + 1.59x4 + 7.47x5 + 2.36x6',
    'constraints': [
        '10x1 <= 283',
        '5x2 <= 283',
        '2x3 <= 283',
        '2x4 <= 283',
        '10x5 <= 283',
        '6x6 <= 283',
        '10x1 + 2x3 >= 47',
        '2x4 + 10x5 >= 47',
        '2x4 + 6x6 >= 47',
        '10x5 + 6x6 >= 38',
        '5x2 + 10x5 >= 27',
        '2x3 + 2x4 >= 37',
        '10x1 + 2x3 + 2x4 >= 37',
        '5x2 + 2x3 + 6x6 >= 37',
        '10x1 + 5x2 + 2x3 >= 37',
        '10x1 + 5x2 + 6x6 >= 37',
        '10x1 + 2x3 + 2x4 >= 38',
        '5x2 + 2x3 + 6x6 >= 38',
        '10x1 + 5x2 + 2x3 >= 38',
        '10x1 + 5x2 + 6x6 >= 38',
        '10x1 + 2x3 + 2x4 >= 37',
        '5x2 + 2x3 + 6x6 >= 37',
        '10x1 + 5x2 + 2x3 >= 37',
        '10x1 + 5x2 + 6x6 >= 37',
        '10x1 + 2x3 + 2x4 >= 29',
        '5x2 + 2x3 + 6x6 >= 29',
        '10x1 + 5x2 + 2x3 >= 29',
        '10x1 + 5x2 + 6x6 >= 29',
        'x2 - 9x5 + 5x6 >= 0',
        '5x2 + 10x5 <= 144',
        '5x2 + 2x4 <= 163',
        '10x1 + 2x4 <= 176',
        '2x3 + 6x6 <= 225',
        '5x2 + 10x5 + 6x6 <= 59',
        '10x1 + 5x2 + 2x4 <= 105',
        '10x1 + 2x3 + 6x6 <= 133',
        '2x3 + 2x4 + 10x5 <= 56',
        '10x1 + 5x2 + 10x5 <= 223',
        '5x2 + 2x4 + 10x5 <= 192',
        '5x2 + 2x3 + 2x4 <= 107',
        '2x3 + 2x4 + 6x6 <= 188',
        '10x1 + 2x3 + 2x4 <= 108',
        '10x1 + 5x2 + 2x3 + 2x4 + 10x5 + 6x6 <= 108'
    ]
}
```

## Step 5: Implement the optimization problem using Gurobi
```python
import gurobi

def optimize_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name='x1', lb=0)  # hours worked by Ringo
    x2 = model.addVar(name='x2', lb=0)  # hours worked by Mary
    x3 = model.addVar(name='x3', lb=0)  # hours worked by Laura
    x4 = model.addVar(name='x4', lb=0)  # hours worked by Jean
    x5 = model.addVar(name='x5', lb=0)  # hours worked by Hank
    x6 = model.addVar(name='x6', lb=0)  # hours worked by John

    # Objective function
    model.setObjective(6.8 * x1 + 4.4 * x2 + 4.92 * x3 + 1.59 * x4 + 7.47 * x5 + 2.36 * x6, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(10 * x1 <= 283)
    model.addConstr(5 * x2 <= 283)
    model.addConstr(2 * x3 <= 283)
    model.addConstr(2 * x4 <= 283)
    model.addConstr(10 * x5 <= 283)
    model.addConstr(6 * x6 <= 283)

    model.addConstr(10 * x1 + 2 * x3 >= 47)
    model.addConstr(2 * x4 + 10 * x5 >= 47)
    model.addConstr(2 * x4 + 6 * x6 >= 47)
    model.addConstr(10 * x5 + 6 * x6 >= 38)
    model.addConstr(5 * x2 + 10 * x5 >= 27)
    model.addConstr(2 * x3 + 2 * x4 >= 37)
    model.addConstr(10 * x1 + 2 * x3 + 2 * x4 >= 37)
    model.addConstr(5 * x2 + 2 * x3 + 6 * x6 >= 37)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x3 >= 37)
    model.addConstr(10 * x1 + 5 * x2 + 6 * x6 >= 37)
    model.addConstr(10 * x1 + 2 * x3 + 2 * x4 >= 38)
    model.addConstr(5 * x2 + 2 * x3 + 6 * x6 >= 38)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x3 >= 38)
    model.addConstr(10 * x1 + 5 * x2 + 6 * x6 >= 38)

    model.addConstr(10 * x1 + 2 * x3 + 2 * x4 >= 37)
    model.addConstr(5 * x2 + 2 * x3 + 6 * x6 >= 37)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x3 >= 37)
    model.addConstr(10 * x1 + 5 * x2 + 6 * x6 >= 37)

    model.addConstr(10 * x1 + 2 * x3 + 2 * x4 >= 29)
    model.addConstr(5 * x2 + 2 * x3 + 6 * x6 >= 29)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x3 >= 29)
    model.addConstr(10 * x1 + 5 * x2 + 6 * x6 >= 29)

    model.addConstr(x2 - 9 * x5 + 5 * x6 >= 0)
    model.addConstr(5 * x2 + 10 * x5 <= 144)
    model.addConstr(5 * x2 + 2 * x4 <= 163)
    model.addConstr(10 * x1 + 2 * x4 <= 176)
    model.addConstr(2 * x3 + 6 * x6 <= 225)
    model.addConstr(5 * x2 + 10 * x5 + 6 * x6 <= 59)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x4 <= 105)
    model.addConstr(10 * x1 + 2 * x3 + 6 * x6 <= 133)
    model.addConstr(2 * x3 + 2 * x4 + 10 * x5 <= 56)
    model.addConstr(10 * x1 + 5 * x2 + 10 * x5 <= 223)
    model.addConstr(5 * x2 + 2 * x4 + 10 * x5 <= 192)
    model.addConstr(5 * x2 + 2 * x3 + 2 * x4 <= 107)
    model.addConstr(2 * x3 + 2 * x4 + 6 * x6 <= 188)
    model.addConstr(10 * x1 + 2 * x3 + 2 * x4 <= 108)
    model.addConstr(10 * x1 + 5 * x2 + 2 * x3 + 2 * x4 + 10 * x5 + 6 * x6 <= 108)

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Ringo: {x1.varValue}")
        print(f"Hours worked by Mary: {x2.varValue}")
        print(f"Hours worked by Laura: {x3.varValue}")
        print(f"Hours worked by Jean: {x4.varValue}")
        print(f"Hours worked by Hank: {x5.varValue}")
        print(f"Hours worked by John: {x6.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

optimize_problem()
```