## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Dale'
- 'x2' corresponds to 'hours worked by Laura'
- 'x3' corresponds to 'hours worked by John'
- 'x4' corresponds to 'hours worked by Peggy'
- 'x5' corresponds to 'hours worked by Hank'

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: 
\[ 1.88x_1 + 7.5x_2 + 9.6x_3 + 3.92x_4 + 6.52x_5 \]

## 3: Define the constraints in symbolic notation
The constraints are:
- \( 13x_1 + 18x_2 + 9x_3 + 17x_4 + 11x_5 \leq 596 \) (total paperwork competence rating)
- \( 18x_2 + 17x_4 \geq 116 \)
- \( 13x_1 + 9x_3 \geq 105 \)
- \( 13x_1 + 11x_5 \geq 88 \)
- \( 18x_2 + 11x_5 \geq 80 \)
- \( 9x_3 + 17x_4 \geq 96 \)
- \( 13x_1 + 18x_2 + 9x_3 \geq 92 \)
- \( 9x_3 + 17x_4 + 11x_5 \geq 92 \)
- \( 13x_1 + 18x_2 + 9x_3 \geq 84 \)
- \( 9x_3 + 17x_4 + 11x_5 \geq 84 \)
- \( 18x_2 + 11x_5 \leq 294 \)
- \( 17x_4 + 11x_5 \leq 265 \)
- \( 18x_2 + 17x_4 \leq 451 \)
- \( 13x_1 + 9x_3 \leq 176 \)
- \( 13x_1 + 18x_2 \leq 481 \)
- \( 13x_1 + 11x_5 \leq 300 \)
- \( 9x_3 + 11x_5 \leq 575 \)
- \( 13x_1 + 18x_2 + 9x_3 + 17x_4 + 11x_5 \leq 575 \)

## 4: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="hours_worked_by_Dale", lb=0)
    x2 = model.addVar(name="hours_worked_by_Laura", lb=0)
    x3 = model.addVar(name="hours_worked_by_John", lb=0)
    x4 = model.addVar(name="hours_worked_by_Peggy", lb=0)
    x5 = model.addVar(name="hours_worked_by_Hank", lb=0)

    # Define the objective function
    model.setObjective(1.88 * x1 + 7.5 * x2 + 9.6 * x3 + 3.92 * x4 + 6.52 * x5, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(13 * x1 + 18 * x2 + 9 * x3 + 17 * x4 + 11 * x5 <= 596)
    model.addConstr(18 * x2 + 17 * x4 >= 116)
    model.addConstr(13 * x1 + 9 * x3 >= 105)
    model.addConstr(13 * x1 + 11 * x5 >= 88)
    model.addConstr(18 * x2 + 11 * x5 >= 80)
    model.addConstr(9 * x3 + 17 * x4 >= 96)
    model.addConstr(13 * x1 + 18 * x2 + 9 * x3 >= 92)
    model.addConstr(9 * x3 + 17 * x4 + 11 * x5 >= 92)
    model.addConstr(13 * x1 + 18 * x2 + 9 * x3 >= 84)
    model.addConstr(9 * x3 + 17 * x4 + 11 * x5 >= 84)
    model.addConstr(18 * x2 + 11 * x5 <= 294)
    model.addConstr(17 * x4 + 11 * x5 <= 265)
    model.addConstr(18 * x2 + 17 * x4 <= 451)
    model.addConstr(13 * x1 + 9 * x3 <= 176)
    model.addConstr(13 * x1 + 18 * x2 <= 481)
    model.addConstr(13 * x1 + 11 * x5 <= 300)
    model.addConstr(9 * x3 + 11 * x5 <= 575)
    model.addConstr(13 * x1 + 18 * x2 + 9 * x3 + 17 * x4 + 11 * x5 <= 575)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Dale: ", x1.varValue)
        print("Hours worked by Laura: ", x2.varValue)
        print("Hours worked by John: ", x3.varValue)
        print("Hours worked by Peggy: ", x4.varValue)
        print("Hours worked by Hank: ", x5.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```

## 5: Symbolic representation of the problem
```json
{
    "sym_variables": [
        ["x1", "hours worked by Dale"],
        ["x2", "hours worked by Laura"],
        ["x3", "hours worked by John"],
        ["x4", "hours worked by Peggy"],
        ["x5", "hours worked by Hank"]
    ],
    "objective_function": "1.88x1 + 7.5x2 + 9.6x3 + 3.92x4 + 6.52x5",
    "constraints": [
        "13x1 + 18x2 + 9x3 + 17x4 + 11x5 <= 596",
        "18x2 + 17x4 >= 116",
        "13x1 + 9x3 >= 105",
        "13x1 + 11x5 >= 88",
        "18x2 + 11x5 >= 80",
        "9x3 + 17x4 >= 96",
        "13x1 + 18x2 + 9x3 >= 92",
        "9x3 + 17x4 + 11x5 >= 92",
        "13x1 + 18x2 + 9x3 >= 84",
        "9x3 + 17x4 + 11x5 >= 84",
        "18x2 + 11x5 <= 294",
        "17x4 + 11x5 <= 265",
        "18x2 + 17x4 <= 451",
        "13x1 + 9x3 <= 176",
        "13x1 + 18x2 <= 481",
        "13x1 + 11x5 <= 300",
        "9x3 + 11x5 <= 575",
        "13x1 + 18x2 + 9x3 + 17x4 + 11x5 <= 575"
    ]
}
```