## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Ringo', 'hours worked by Mary', 'hours worked by George', 'hours worked by Peggy', 'hours worked by Hank']. Let's denote them as $x_0, x_1, x_2, x_3, x_4$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $4x_0 + 6x_1 + 4x_2 + 3x_3 + 1x_4$.

## Step 3: List all the constraints
1. $3x_2 + 3x_4 \geq 8$
2. $1x_0 + 2x_3 \geq 5$
3. $4x_1 + 2x_4 \geq 6$
4. $1x_0 + 2x_2 \geq 11$
5. $1x_0 + 5x_3 \geq 5$
6. $2x_2 + 5x_3 + 2x_4 \geq 11$
7. $3x_2 + 2x_3 \leq 82$
8. $5x_0 + 3x_2 \leq 31$
9. $4x_1 + 2x_3 \leq 61$
10. $5x_0 + 2x_3 \leq 41$
11. $5x_0 + 4x_1 \leq 100$
12. $5x_0 + 3x_4 \leq 52$
13. $4x_1 + 3x_2 \leq 66$
14. $4x_1 + 3x_2 + 2x_3 \leq 41$
15. $5x_0 + 4x_1 + 2x_3 \leq 34$
16. $5x_0 + 4x_1 + 3x_2 + 2x_3 + 3x_4 \leq 34$
17. $2x_2 + 2x_4 \leq 39$
18. $1x_0 + 4x_1 \leq 45$
19. $4x_1 + 2x_4 \leq 43$
20. $2x_3 + 2x_4 \leq 31$
21. $1x_0 + 4x_1 + 2x_4 \leq 45$
22. $1x_0 + 4x_1 + 2x_2 + 2x_3 + 2x_4 \leq 45$
23. $5x_1 + 2x_2 \leq 40$
24. $5x_1 + 2x_4 \leq 44$
25. $1x_0 + 5x_1 \leq 34$
26. $1x_0 + 5x_1 + 2x_2 + 5x_3 + 2x_4 \leq 34$

## 4: Define the symbolic representation of the problem in JSON format
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Ringo'), 
        ('x1', 'hours worked by Mary'), 
        ('x2', 'hours worked by George'), 
        ('x3', 'hours worked by Peggy'), 
        ('x4', 'hours worked by Hank')
    ], 
    'objective_function': '4*x0 + 6*x1 + 4*x2 + 3*x3 + 1*x4', 
    'constraints': [
        '3*x2 + 3*x4 >= 8',
        '1*x0 + 2*x3 >= 5',
        '4*x1 + 2*x4 >= 6',
        '1*x0 + 2*x2 >= 11',
        '1*x0 + 5*x3 >= 5',
        '2*x2 + 5*x3 + 2*x4 >= 11',
        '3*x2 + 2*x3 <= 82',
        '5*x0 + 3*x2 <= 31',
        '4*x1 + 2*x3 <= 61',
        '5*x0 + 2*x3 <= 41',
        '5*x0 + 4*x1 <= 100',
        '5*x0 + 3*x4 <= 52',
        '4*x1 + 3*x2 <= 66',
        '4*x1 + 3*x2 + 2*x3 <= 41',
        '5*x0 + 4*x1 + 2*x3 <= 34',
        '5*x0 + 4*x1 + 3*x2 + 2*x3 + 3*x4 <= 34',
        '2*x2 + 2*x4 <= 39',
        '1*x0 + 4*x1 <= 45',
        '4*x1 + 2*x4 <= 43',
        '2*x3 + 2*x4 <= 31',
        '1*x0 + 4*x1 + 2*x4 <= 45',
        '1*x0 + 4*x1 + 2*x2 + 2*x3 + 2*x4 <= 45',
        '5*x1 + 2*x2 <= 40',
        '5*x1 + 2*x4 <= 44',
        '1*x0 + 5*x1 <= 34',
        '1*x0 + 5*x1 + 2*x2 + 5*x3 + 2*x4 <= 34'
    ]
}
```

## 5: Write the Gurobi code
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x0 = model.addVar(name='x0', lb=0, integrality=1) # hours worked by Ringo
x1 = model.addVar(name='x1', lb=0) # hours worked by Mary
x2 = model.addVar(name='x2', lb=0) # hours worked by George
x3 = model.addVar(name='x3', lb=0) # hours worked by Peggy
x4 = model.addVar(name='x4', lb=0) # hours worked by Hank

# Define the objective function
model.setObjective(4*x0 + 6*x1 + 4*x2 + 3*x3 + 1*x4, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(3*x2 + 3*x4 >= 8)
model.addConstr(1*x0 + 2*x3 >= 5)
model.addConstr(4*x1 + 2*x4 >= 6)
model.addConstr(1*x0 + 2*x2 >= 11)
model.addConstr(1*x0 + 5*x3 >= 5)
model.addConstr(2*x2 + 5*x3 + 2*x4 >= 11)
model.addConstr(3*x2 + 2*x3 <= 82)
model.addConstr(5*x0 + 3*x2 <= 31)
model.addConstr(4*x1 + 2*x3 <= 61)
model.addConstr(5*x0 + 2*x3 <= 41)
model.addConstr(5*x0 + 4*x1 <= 100)
model.addConstr(5*x0 + 3*x4 <= 52)
model.addConstr(4*x1 + 3*x2 <= 66)
model.addConstr(4*x1 + 3*x2 + 2*x3 <= 41)
model.addConstr(5*x0 + 4*x1 + 2*x3 <= 34)
model.addConstr(5*x0 + 4*x1 + 3*x2 + 2*x3 + 3*x4 <= 34)
model.addConstr(2*x2 + 2*x4 <= 39)
model.addConstr(1*x0 + 4*x1 <= 45)
model.addConstr(4*x1 + 2*x4 <= 43)
model.addConstr(2*x3 + 2*x4 <= 31)
model.addConstr(1*x0 + 4*x1 + 2*x4 <= 45)
model.addConstr(1*x0 + 4*x1 + 2*x2 + 2*x3 + 2*x4 <= 45)
model.addConstr(5*x1 + 2*x2 <= 40)
model.addConstr(5*x1 + 2*x4 <= 44)
model.addConstr(1*x0 + 5*x1 <= 34)
model.addConstr(1*x0 + 5*x1 + 2*x2 + 5*x3 + 2*x4 <= 34)

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', model.objVal)
    print('x0: ', x0.varValue)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
else:
    print('No solution found')
```