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

## Step 2: Convert the given problem into a symbolic representation
The objective function to maximize is $1 \cdot x_0 + 1 \cdot x_1 + 6 \cdot x_2$.

## 3: List the constraints in symbolic notation
The constraints given are:
1. $1 \cdot x_0 + 16 \cdot x_1 + 15 \cdot x_2 \geq 64$
2. $6 \cdot x_0 - 4 \cdot x_2 \geq 0$
3. $31 \cdot x_0 + 1 \cdot x_2 \leq 227$
4. $31 \cdot x_0 + 24 \cdot x_1 + 1 \cdot x_2 \leq 227$
5. $1 \cdot x_0 + 15 \cdot x_2 \leq 151$
6. $1 \cdot x_0 + 16 \cdot x_1 \leq 182$
7. $1 \cdot x_0 + 16 \cdot x_1 + 15 \cdot x_2 \leq 162$
8. $x_0$ is an integer
9. $x_1$ is continuous
10. $x_2$ is an integer

## 4: Create the symbolic representation dictionary
The symbolic representation is:
```json
{
'sym_variables': [('x0', 'hours worked by Dale'), ('x1', 'hours worked by Paul'), ('x2', 'hours worked by Laura')],
'objective_function': 'x0 + x1 + 6*x2',
'constraints': [
    'x0 + 16*x1 + 15*x2 >= 64',
    '6*x0 - 4*x2 >= 0',
    '31*x0 + x2 <= 227',
    '31*x0 + 24*x1 + x2 <= 227',
    'x0 + 15*x2 <= 151',
    'x0 + 16*x1 <= 182',
    'x0 + 16*x1 + 15*x2 <= 162'
]
}
```

## 5: Write the Gurobi code for the optimization problem
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name='hours_worked_by_Dale', vtype='I')  # Integer
    x1 = model.addVar(name='hours_worked_by_Paul', vtype='C')  # Continuous
    x2 = model.addVar(name='hours_worked_by_Laura', vtype='I')  # Integer

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

    # Add constraints
    model.addConstr(x0 + 16 * x1 + 15 * x2 >= 64)
    model.addConstr(6 * x0 - 4 * x2 >= 0)
    model.addConstr(31 * x0 + x2 <= 227)
    model.addConstr(31 * x0 + 24 * x1 + x2 <= 227)
    model.addConstr(x0 + 15 * x2 <= 151)
    model.addConstr(x0 + 16 * x1 <= 182)
    model.addConstr(x0 + 16 * x1 + 15 * x2 <= 162)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Optimal solution found.')
        print(f'Hours worked by Dale: {x0.varValue}')
        print(f'Hours worked by Paul: {x1.varValue}')
        print(f'Hours worked by Laura: {x2.varValue}')
        print(f'Objective function value: {model.objVal}')
    else:
        print('No optimal solution found.')

solve_optimization_problem()
```