## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Peggy', 'hours worked by Paul', and 'hours worked by Bill'. Let's denote them as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language objects into symbolic variables
The symbolic representation of the variables is:
- $x_1$ for 'hours worked by Peggy'
- $x_2$ for 'hours worked by Paul'
- $x_3$ for 'hours worked by Bill'

## 3: Define the objective function in symbolic notation
The objective function to maximize is $8.96x_1 + 1.78x_2 + 9.49x_3$.

## 4: Define the constraints in symbolic notation
The constraints are:
- $20x_1 \geq 0$ (implicit, as $x_1$ represents hours and is non-negative)
- $17x_2 \geq 0$ (implicit, as $x_2$ represents hours and is non-negative)
- $4x_3 \geq 0$ (implicit, as $x_3$ represents hours and is non-negative)
- $20x_1 + 17x_2 \geq 38$
- $20x_1 + 4x_3 \geq 35$
- $17x_2 + 4x_3 \geq 50$
- $20x_1 + 17x_2 + 4x_3 \geq 83$
- $17x_2 + 4x_3 \leq 198$
- $20x_1 + 17x_2 \leq 166$
- $20x_1 + 17x_2 + 4x_3 \leq 166$

## 5: Create the symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Peggy'), ('x2', 'hours worked by Paul'), ('x3', 'hours worked by Bill')],
    'objective_function': '8.96*x1 + 1.78*x2 + 9.49*x3',
    'constraints': [
        '20*x1 + 17*x2 >= 38',
        '20*x1 + 4*x3 >= 35',
        '17*x2 + 4*x3 >= 50',
        '20*x1 + 17*x2 + 4*x3 >= 83',
        '17*x2 + 4*x3 <= 198',
        '20*x1 + 17*x2 <= 166',
        '20*x1 + 17*x2 + 4*x3 <= 166'
    ]
}
```

## 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="hours_worked_by_Peggy", lb=0)  # Assuming hours cannot be negative
    x2 = model.addVar(name="hours_worked_by_Paul", lb=0)   # Assuming hours cannot be negative
    x3 = model.addVar(name="hours_worked_by_Bill", lb=0)   # Assuming hours cannot be negative

    # Define the objective function
    model.setObjective(8.96 * x1 + 1.78 * x2 + 9.49 * x3, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(20 * x1 + 17 * x2 >= 38)
    model.addConstr(20 * x1 + 4 * x3 >= 35)
    model.addConstr(17 * x2 + 4 * x3 >= 50)
    model.addConstr(20 * x1 + 17 * x2 + 4 * x3 >= 83)
    model.addConstr(17 * x2 + 4 * x3 <= 198)
    model.addConstr(20 * x1 + 17 * x2 <= 166)
    model.addConstr(20 * x1 + 17 * x2 + 4 * x3 <= 166)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Peggy: {x1.varValue}")
        print(f"Hours worked by Paul: {x2.varValue}")
        print(f"Hours worked by Bill: {x3.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```