## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Ringo' and 'hours worked by Bill'. Let's denote 'hours worked by Ringo' as $x_1$ and 'hours worked by Bill' as $x_2$. The objective function to maximize is $3.3x_1 + 8.28x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $18x_1 + 3x_2 \geq 38$
2. $32x_1 + 18x_2 \geq 45$
3. $9x_1 - 2x_2 \geq 0$
4. $18x_1 + 3x_2 \leq 62$
5. $32x_1 + 18x_2 \leq 84$
6. $x_1$ can be any real number (including fractions)
7. $x_2$ must be an integer

## 3: Convert the problem into a Gurobi-compatible format
To solve this problem using Gurobi, we need to define the model, add the variables, the objective function, and the constraints.

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

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name="hours_worked_by_Ringo", lb=0)  # No lower bound specified, assuming 0
x2 = model.addVar(name="hours_worked_by_Bill", lb=0, integrality=gurobi.GRB.INTEGER)  # Bill's hours must be an integer

# Define the objective function
model.setObjective(3.3 * x1 + 8.28 * x2, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(18 * x1 + 3 * x2 >= 38, name="paperwork_competence_rating")
model.addConstr(32 * x1 + 18 * x2 >= 45, name="productivity_rating")
model.addConstr(9 * x1 - 2 * x2 >= 0, name="productivity_vs_paperwork")
model.addConstr(18 * x1 + 3 * x2 <= 62, name="max_paperwork_competence_rating")
model.addConstr(32 * x1 + 18 * x2 <= 84, name="max_productivity_rating")

# Optimize the model
model.optimize()

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

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [('x1', 'hours worked by Ringo'), ('x2', 'hours worked by Bill')],
    'objective_function': '3.3*x1 + 8.28*x2',
    'constraints': [
        '18*x1 + 3*x2 >= 38',
        '32*x1 + 18*x2 >= 45',
        '9*x1 - 2*x2 >= 0',
        '18*x1 + 3*x2 <= 62',
        '32*x1 + 18*x2 <= 84'
    ]
}
```

The final answer is: 

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(name="hours_worked_by_Ringo", lb=0)  
x2 = model.addVar(name="hours_worked_by_Bill", lb=0, integrality=gurobi.GRB.INTEGER)  

# Define the objective function
model.setObjective(3.3 * x1 + 8.28 * x2, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(18 * x1 + 3 * x2 >= 38, name="paperwork_competence_rating")
model.addConstr(32 * x1 + 18 * x2 >= 45, name="productivity_rating")
model.addConstr(9 * x1 - 2 * x2 >= 0, name="productivity_vs_paperwork")
model.addConstr(18 * x1 + 3 * x2 <= 62, name="max_paperwork_competence_rating")
model.addConstr(32 * x1 + 18 * x2 <= 84, name="max_productivity_rating")

# Optimize the model
model.optimize()

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