## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Bill', 'hours worked by Peggy', and 'hours worked by Jean', 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 $8x_0 + 9x_1 + 4x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $16x_1 + 5x_2 \geq 51$
- $12x_0 + 5x_2 \geq 24$
- $20x_0 + 12x_2 \geq 34$
- $16x_1 + 12x_2 \geq 96$
- $18x_0 + 2x_2 \geq 42$
- $18x_0 + 11x_1 + 2x_2 \geq 37$
- $16x_1 + 5x_2 \leq 118$
- $12x_0 + 5x_2 \leq 163$
- $12x_0 + 16x_1 + 5x_2 \leq 125$
- $20x_0 + 12x_2 \leq 194$
- $20x_0 + 16x_1 + 12x_2 \leq 194$
- $11x_1 + 2x_2 \leq 134$
- $18x_0 + 11x_1 \leq 64$
- $18x_0 + 11x_1 + 2x_2 \leq 64$

## 4: Create a symbolic representation of the problem
```json
{
    'sym_variables': [('x0', 'hours worked by Bill'), ('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Jean')],
    'objective_function': '8*x0 + 9*x1 + 4*x2',
    'constraints': [
        '16*x1 + 5*x2 >= 51',
        '12*x0 + 5*x2 >= 24',
        '20*x0 + 12*x2 >= 34',
        '16*x1 + 12*x2 >= 96',
        '18*x0 + 2*x2 >= 42',
        '18*x0 + 11*x1 + 2*x2 >= 37',
        '16*x1 + 5*x2 <= 118',
        '12*x0 + 5*x2 <= 163',
        '12*x0 + 16*x1 + 5*x2 <= 125',
        '20*x0 + 12*x2 <= 194',
        '20*x0 + 16*x1 + 12*x2 <= 194',
        '11*x1 + 2*x2 <= 134',
        '18*x0 + 11*x1 <= 64',
        '18*x0 + 11*x1 + 2*x2 <= 64'
    ]
}
```

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

# Create a new model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="x0", lb=0)  # hours worked by Bill
x1 = m.addVar(name="x1", lb=0)  # hours worked by Peggy
x2 = m.addVar(name="x2", lb=0)  # hours worked by Jean

# Define the objective function
m.setObjective(8*x0 + 9*x1 + 4*x2, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(16*x1 + 5*x2 >= 51)
m.addConstr(12*x0 + 5*x2 >= 24)
m.addConstr(20*x0 + 12*x2 >= 34)
m.addConstr(16*x1 + 12*x2 >= 96)
m.addConstr(18*x0 + 2*x2 >= 42)
m.addConstr(18*x0 + 11*x1 + 2*x2 >= 37)
m.addConstr(16*x1 + 5*x2 <= 118)
m.addConstr(12*x0 + 5*x2 <= 163)
m.addConstr(12*x0 + 16*x1 + 5*x2 <= 125)
m.addConstr(20*x0 + 12*x2 <= 194)
m.addConstr(20*x0 + 16*x1 + 12*x2 <= 194)
m.addConstr(11*x1 + 2*x2 <= 134)
m.addConstr(18*x0 + 11*x1 <= 64)
m.addConstr(18*x0 + 11*x1 + 2*x2 <= 64)

# Solve the model
m.optimize()

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