## Step 1: Define the symbolic representation of the problem
The problem involves maximizing an objective function subject to several constraints. The variables are 'hours worked by Bill', 'hours worked by Mary', and 'hours worked by Hank', which can be represented symbolically as $x_0$, $x_1$, and $x_2$ respectively.

## Step 2: Identify the objective function
The objective function to maximize is $9.62x_0 + 6.74x_1 + 7.61x_2$.

## 3: List the constraints
The constraints given are:
- $4.81x_0 + 10.91x_1 \geq 18$
- $4.81x_0 + 10.91x_1 + 7.87x_2 \geq 18$
- $10.28x_0 + 0.89x_1 + 11.33x_2 \geq 34$
- $0.62x_0 + 8.79x_2 \geq 14$
- $11.77x_1 + 8.79x_2 \geq 37$
- $5.35x_0 + 8.39x_1 + 11.28x_2 \geq 28$
- $11.19x_1 + 9.05x_2 \leq 75$
- $9.76x_0 + 11.19x_1 + 9.05x_2 \leq 75$
- $4.81x_0 + 7.87x_2 \leq 51$
- $4.81x_0 + 10.91x_1 + 7.87x_2 \leq 51$
- $10.28x_0 + 11.33x_2 \leq 83$
- $0.89x_1 + 11.33x_2 \leq 64$
- $10.28x_0 + 0.89x_1 + 11.33x_2 \leq 64$
- $0.62x_0 + 8.79x_2 \leq 38$
- $11.77x_1 + 8.79x_2 \leq 86$
- $0.62x_0 + 11.77x_1 + 8.79x_2 \leq 86$
- $5.35x_0 + 8.39x_1 \leq 105$
- $5.35x_0 + 11.28x_2 \leq 110$
- $5.35x_0 + 8.39x_1 + 11.28x_2 \leq 110$

## 4: Create a symbolic representation of the problem
The symbolic representation is:
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Bill'),
        ('x1', 'hours worked by Mary'),
        ('x2', 'hours worked by Hank')
    ],
    'objective_function': '9.62*x0 + 6.74*x1 + 7.61*x2',
    'constraints': [
        '4.81*x0 + 10.91*x1 >= 18',
        '4.81*x0 + 10.91*x1 + 7.87*x2 >= 18',
        '10.28*x0 + 0.89*x1 + 11.33*x2 >= 34',
        '0.62*x0 + 8.79*x2 >= 14',
        '11.77*x1 + 8.79*x2 >= 37',
        '5.35*x0 + 8.39*x1 + 11.28*x2 >= 28',
        '11.19*x1 + 9.05*x2 <= 75',
        '9.76*x0 + 11.19*x1 + 9.05*x2 <= 75',
        '4.81*x0 + 7.87*x2 <= 51',
        '4.81*x0 + 10.91*x1 + 7.87*x2 <= 51',
        '10.28*x0 + 11.33*x2 <= 83',
        '0.89*x1 + 11.33*x2 <= 64',
        '10.28*x0 + 0.89*x1 + 11.33*x2 <= 64',
        '0.62*x0 + 8.79*x2 <= 38',
        '11.77*x1 + 8.79*x2 <= 86',
        '0.62*x0 + 11.77*x1 + 8.79*x2 <= 86',
        '5.35*x0 + 8.39*x1 <= 105',
        '5.35*x0 + 11.28*x2 <= 110',
        '5.35*x0 + 8.39*x1 + 11.28*x2 <= 110'
    ]
}
```

## 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="x0", lb=0)  # hours worked by Bill
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Mary
    x2 = model.addVar(name="x2", lb=0)  # hours worked by Hank

    # Define the objective function
    model.setObjective(9.62 * x0 + 6.74 * x1 + 7.61 * x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(4.81 * x0 + 10.91 * x1 >= 18)
    model.addConstr(4.81 * x0 + 10.91 * x1 + 7.87 * x2 >= 18)
    model.addConstr(10.28 * x0 + 0.89 * x1 + 11.33 * x2 >= 34)
    model.addConstr(0.62 * x0 + 8.79 * x2 >= 14)
    model.addConstr(11.77 * x1 + 8.79 * x2 >= 37)
    model.addConstr(5.35 * x0 + 8.39 * x1 + 11.28 * x2 >= 28)
    model.addConstr(11.19 * x1 + 9.05 * x2 <= 75)
    model.addConstr(9.76 * x0 + 11.19 * x1 + 9.05 * x2 <= 75)
    model.addConstr(4.81 * x0 + 7.87 * x2 <= 51)
    model.addConstr(4.81 * x0 + 10.91 * x1 + 7.87 * x2 <= 51)
    model.addConstr(10.28 * x0 + 11.33 * x2 <= 83)
    model.addConstr(0.89 * x1 + 11.33 * x2 <= 64)
    model.addConstr(10.28 * x0 + 0.89 * x1 + 11.33 * x2 <= 64)
    model.addConstr(0.62 * x0 + 8.79 * x2 <= 38)
    model.addConstr(11.77 * x1 + 8.79 * x2 <= 86)
    model.addConstr(0.62 * x0 + 11.77 * x1 + 8.79 * x2 <= 86)
    model.addConstr(5.35 * x0 + 8.39 * x1 <= 105)
    model.addConstr(5.35 * x0 + 11.28 * x2 <= 110)
    model.addConstr(5.35 * x0 + 8.39 * x1 + 11.28 * x2 <= 110)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Bill: ", x0.varValue)
        print("Hours worked by Mary: ", x1.varValue)
        print("Hours worked by Hank: ", x2.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```