## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Jean', 'hours worked by Bobby', and 'hours worked by John', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $6x_0 + 9x_1 + 1x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $11.52x_0 + 4.15x_1 + 10.69x_2 \geq 37$
- $0.66x_0 + 12.98x_1 + 0.6x_2 \geq 83$
- $3.8x_0 + 2.18x_1 \geq 19$
- $4.15x_1 + 10.69x_2 \leq 78$
- $11.52x_0 + 4.15x_1 \leq 134$
- $11.52x_0 + 4.15x_1 + 10.69x_2 \leq 134$
- $12.98x_1 + 0.6x_2 \leq 239$
- $0.66x_0 + 0.6x_2 \leq 134$
- $0.66x_0 + 12.98x_1 + 0.6x_2 \leq 219$
- $3.8x_0 + 2.18x_1 \leq 61$
- $2.18x_1 + 4.38x_2 \leq 90$
- $3.8x_0 + 2.18x_1 + 4.38x_2 \leq 89$

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Jean'), 
        ('x1', 'hours worked by Bobby'), 
        ('x2', 'hours worked by John')
    ], 
    'objective_function': '6*x0 + 9*x1 + 1*x2', 
    'constraints': [
        '11.52*x0 + 4.15*x1 + 10.69*x2 >= 37',
        '0.66*x0 + 12.98*x1 + 0.6*x2 >= 83',
        '3.8*x0 + 2.18*x1 >= 19',
        '4.15*x1 + 10.69*x2 <= 78',
        '11.52*x0 + 4.15*x1 <= 134',
        '11.52*x0 + 4.15*x1 + 10.69*x2 <= 134',
        '12.98*x1 + 0.6*x2 <= 239',
        '0.66*x0 + 0.6*x2 <= 134',
        '0.66*x0 + 12.98*x1 + 0.6*x2 <= 219',
        '3.8*x0 + 2.18*x1 <= 61',
        '2.18*x1 + 4.38*x2 <= 90',
        '3.8*x0 + 2.18*x1 + 4.38*x2 <= 89'
    ]
}
```

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

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define the variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Jean
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Bobby
    x2 = model.addVar(name="x2", lb=0)  # hours worked by John

    # Define the objective function
    model.setObjective(6 * x0 + 9 * x1 + 1 * x2, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(11.52 * x0 + 4.15 * x1 + 10.69 * x2 >= 37)
    model.addConstr(0.66 * x0 + 12.98 * x1 + 0.6 * x2 >= 83)
    model.addConstr(3.8 * x0 + 2.18 * x1 >= 19)
    model.addConstr(4.15 * x1 + 10.69 * x2 <= 78)
    model.addConstr(11.52 * x0 + 4.15 * x1 <= 134)
    model.addConstr(11.52 * x0 + 4.15 * x1 + 10.69 * x2 <= 134)
    model.addConstr(12.98 * x1 + 0.6 * x2 <= 239)
    model.addConstr(0.66 * x0 + 0.6 * x2 <= 134)
    model.addConstr(0.66 * x0 + 12.98 * x1 + 0.6 * x2 <= 219)
    model.addConstr(3.8 * x0 + 2.18 * x1 <= 61)
    model.addConstr(2.18 * x1 + 4.38 * x2 <= 90)
    model.addConstr(3.8 * x0 + 2.18 * x1 + 4.38 * x2 <= 89)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Jean: ", x0.varValue)
        print("Hours worked by Bobby: ", x1.varValue)
        print("Hours worked by John: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```