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

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $6x_0 + 4x_1$.

## Step 3: List the constraints
The constraints given are:
- $13x_0 + 3x_1 \geq 8$ (organization score)
- $6x_0 + 14x_1 \geq 29$ (likelihood to quit index)
- $6x_0 + 13x_1 \geq 7$ (dollar cost per hour)
- $11x_0 + 11x_1 \geq 17$ (paperwork competence rating)
- $4x_0 - 7x_1 \geq 0$
- $13x_0 + 3x_1 \leq 42$ (organization score upper bound)
- $6x_0 + 14x_1 \leq 42$ (likelihood to quit index upper bound)
- $6x_0 + 13x_1 \leq 38$ (dollar cost per hour upper bound)
- $11x_0 + 11x_1 \leq 20$ (paperwork competence rating upper bound)

## 4: Create a symbolic representation of the problem
The symbolic representation is:
```json
{
'sym_variables': [('x0', 'hours worked by Paul'), ('x1', 'hours worked by Hank')],
'objective_function': '6*x0 + 4*x1',
'constraints': [
    '13*x0 + 3*x1 >= 8',
    '6*x0 + 14*x1 >= 29',
    '6*x0 + 13*x1 >= 7',
    '11*x0 + 11*x1 >= 17',
    '4*x0 - 7*x1 >= 0',
    '13*x0 + 3*x1 <= 42',
    '6*x0 + 14*x1 <= 42',
    '6*x0 + 13*x1 <= 38',
    '11*x0 + 11*x1 <= 20'
]
}
```

## 5: Write the Gurobi code
```python
import gurobi

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

    # Define the variables
    x0 = model.addVar(name="hours_worked_by_Paul", lb=0)  # hours worked by Paul
    x1 = model.addVar(name="hours_worked_by_Hank", lb=0)  # hours worked by Hank

    # Define the objective function
    model.setObjective(6 * x0 + 4 * x1, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(13 * x0 + 3 * x1 >= 8, name="organization_score")
    model.addConstr(6 * x0 + 14 * x1 >= 29, name="likelihood_to_quit_index")
    model.addConstr(6 * x0 + 13 * x1 >= 7, name="dollar_cost_per_hour")
    model.addConstr(11 * x0 + 11 * x1 >= 17, name="paperwork_competence_rating")
    model.addConstr(4 * x0 - 7 * x1 >= 0, name="ratio_constraint")
    model.addConstr(13 * x0 + 3 * x1 <= 42, name="organization_score_upper_bound")
    model.addConstr(6 * x0 + 14 * x1 <= 42, name="likelihood_to_quit_index_upper_bound")
    model.addConstr(6 * x0 + 13 * x1 <= 38, name="dollar_cost_per_hour_upper_bound")
    model.addConstr(11 * x0 + 11 * x1 <= 20, name="paperwork_competence_rating_upper_bound")

    # Optimize the model
    model.optimize()

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

solve_optimization_problem()
```