## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'hours worked by Paul' and 'hours worked by Dale'. Let's denote 'hours worked by Paul' as $x_1$ and 'hours worked by Dale' as $x_2$. The objective function to maximize is $3.56x_1 + 9.21x_2$.

## Step 2: List the constraints in symbolic notation
The constraints given are:
1. $0.44x_1 + 0.93x_2 \geq 21$
2. $0.47x_1 + 0.89x_2 \geq 12$
3. $8x_1 - 5x_2 \geq 0$
4. $0.44x_1 + 0.93x_2 \leq 38$
5. $0.47x_1 + 0.89x_2 \leq 21$

## 3: Define the symbolic variables and objective function
The symbolic variables are:
- $x_1$ for 'hours worked by Paul'
- $x_2$ for 'hours worked by Dale'

The objective function is $3.56x_1 + 9.21x_2$.

## 4: Express the problem in a JSON format for clarity
```json
{
'sym_variables': [('x1', 'hours worked by Paul'), ('x2', 'hours worked by Dale')],
'objective_function': '3.56*x1 + 9.21*x2',
'constraints': [
    '0.44*x1 + 0.93*x2 >= 21',
    '0.47*x1 + 0.89*x2 >= 12',
    '8*x1 - 5*x2 >= 0',
    '0.44*x1 + 0.93*x2 <= 38',
    '0.47*x1 + 0.89*x2 <= 21'
]
}
```

## 5: Implement the problem using Gurobi
To solve this problem using Gurobi, we will use Python. First, ensure you have Gurobi installed in your environment.

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="hours_worked_by_Paul", lb=0)  # Lower bound is 0, assuming hours cannot be negative
    x2 = model.addVar(name="hours_worked_by_Dale", lb=0)  # Lower bound is 0, assuming hours cannot be negative

    # Objective function: maximize 3.56*x1 + 9.21*x2
    model.setObjective(3.56 * x1 + 9.21 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(0.44 * x1 + 0.93 * x2 >= 21, name="paperwork_competence_rating_min")
    model.addConstr(0.47 * x1 + 0.89 * x2 >= 12, name="dollar_cost_per_hour_min")
    model.addConstr(8 * x1 - 5 * x2 >= 0, name="hourly_work_constraint")
    model.addConstr(0.44 * x1 + 0.93 * x2 <= 38, name="paperwork_competence_rating_max")
    model.addConstr(0.47 * x1 + 0.89 * x2 <= 21, name="dollar_cost_per_hour_max")

    # Optimize
    model.optimize()

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

solve_optimization_problem()
```