## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Ringo'
- 'x2' corresponds to 'hours worked by Bobby'
- 'x3' corresponds to 'hours worked by George'
- 'x4' corresponds to 'hours worked by Laura'

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: $1.82x_1 + 5.26x_2 + 3.54x_3 + 6.15x_4$

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $11x_1 \leq 274$ (Implicit upper bound, not directly stated but inferred from 'r0' description)
- $10x_2 \leq 274$
- $1x_3 \leq 274$
- $15x_4 \leq 274$
- $10x_2 + 15x_4 \geq 56$
- $11x_1 + 10x_2 \geq 65$
- $1x_3 + 15x_4 \geq 40$
- $10x_2 + 1x_3 + 15x_4 \geq 44$
- $11x_1 + 1x_3 + 15x_4 \geq 44$
- $10x_2 + 1x_3 + 15x_4 \geq 34$
- $11x_1 + 10x_2 + 1x_3 \geq 34$
- $11x_1 + 1x_3 + 15x_4 \geq 34$
- $10x_2 + 1x_3 + 15x_4 \geq 57$
- $11x_1 + 10x_2 + 1x_3 \geq 57$
- $11x_1 + 1x_3 + 15x_4 \geq 57$
- $1x_3 + 15x_4 \leq 250$
- $11x_1 + 10x_2 \leq 244$
- $10x_2 + 1x_3 \leq 151$
- $11x_1 + 10x_2 + 1x_3 + 15x_4 \leq 151$

## 4: Formulate the problem in Gurobi

```python
import gurobi

def optimization_problem():
    # Create a new model
    m = gurobi.Model()

    # Define the variables
    x1 = m.addVar(name="x1", lb=0)  # hours worked by Ringo
    x2 = m.addVar(name="x2", lb=0)  # hours worked by Bobby
    x3 = m.addVar(name="x3", lb=0, integrality=gurobi.GRB.INTEGER)  # hours worked by George
    x4 = m.addVar(name="x4", lb=0)  # hours worked by Laura

    # Objective function
    m.setObjective(1.82*x1 + 5.26*x2 + 3.54*x3 + 6.15*x4, gurobi.GRB.MAXIMIZE)

    # Constraints
    m.addConstr(11*x1 <= 274)
    m.addConstr(10*x2 <= 274)
    m.addConstr(x3 <= 274)
    m.addConstr(15*x4 <= 274)
    m.addConstr(10*x2 + 15*x4 >= 56)
    m.addConstr(11*x1 + 10*x2 >= 65)
    m.addConstr(x3 + 15*x4 >= 40)
    m.addConstr(10*x2 + x3 + 15*x4 >= 44)
    m.addConstr(11*x1 + x3 + 15*x4 >= 44)
    m.addConstr(10*x2 + x3 + 15*x4 >= 34)
    m.addConstr(11*x1 + 10*x2 + x3 >= 34)
    m.addConstr(11*x1 + x3 + 15*x4 >= 34)
    m.addConstr(10*x2 + x3 + 15*x4 >= 57)
    m.addConstr(11*x1 + 10*x2 + x3 >= 57)
    m.addConstr(11*x1 + x3 + 15*x4 >= 57)
    m.addConstr(x3 + 15*x4 <= 250)
    m.addConstr(11*x1 + 10*x2 <= 244)
    m.addConstr(10*x2 + x3 <= 151)
    m.addConstr(11*x1 + 10*x2 + x3 + 15*x4 <= 151)

    # Optimize the model
    m.optimize()

    # Print the solution
    if m.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", m.objVal)
        print("Hours worked by Ringo: ", x1.varValue)
        print("Hours worked by Bobby: ", x2.varValue)
        print("Hours worked by George: ", x3.varValue)
        print("Hours worked by Laura: ", x4.varValue)
    else:
        print("The model is infeasible")

optimization_problem()
```

## 5: Symbolic Representation

```json
{
    'sym_variables': [('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Bobby'), ('x3', 'hours worked by George'), ('x4', 'hours worked by Laura')],
    'objective_function': '1.82*x1 + 5.26*x2 + 3.54*x3 + 6.15*x4',
    'constraints': [
        '11*x1 <= 274',
        '10*x2 <= 274',
        'x3 <= 274',
        '15*x4 <= 274',
        '10*x2 + 15*x4 >= 56',
        '11*x1 + 10*x2 >= 65',
        'x3 + 15*x4 >= 40',
        '10*x2 + x3 + 15*x4 >= 44',
        '11*x1 + x3 + 15*x4 >= 44',
        '10*x2 + x3 + 15*x4 >= 34',
        '11*x1 + 10*x2 + x3 >= 34',
        '11*x1 + x3 + 15*x4 >= 34',
        '10*x2 + x3 + 15*x4 >= 57',
        '11*x1 + 10*x2 + x3 >= 57',
        '11*x1 + x3 + 15*x4 >= 57',
        'x3 + 15*x4 <= 250',
        '11*x1 + 10*x2 <= 244',
        '10*x2 + x3 <= 151',
        '11*x1 + 10*x2 + x3 + 15*x4 <= 151'
    ]
}
```