## Step 1: Define the symbolic representation of the variables
The variables are defined as follows: 
- 'x1' corresponds to 'hours worked by Hank'
- 'x2' corresponds to 'hours worked by Dale'
- 'x3' corresponds to 'hours worked by Laura'
- 'x4' corresponds to 'hours worked by George'
- 'x5' corresponds to 'hours worked by Peggy'
- 'x6' corresponds to 'hours worked by Paul'

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: $8x_1 + 2x_2 + 4x_3 + 4x_4 + 7x_5 + x_6$

## Step 3: List the constraints in symbolic notation
The constraints are:
- $2x_1 \leq 359$
- $5x_2 \leq 359$
- $16x_3 \leq 359$
- $16x_4 \leq 359$
- $6x_5 \leq 359$
- $9x_6 \leq 359$
- $5x_2 + 6x_5 \geq 31$
- $6x_5 + 9x_6 \geq 42$
- $16x_3 + 9x_6 \geq 33$
- $16x_3 + 6x_5 \geq 33$
- $5x_2 + 16x_4 + 9x_6 \geq 41$
- $2x_1 + 16x_3 + 9x_6 \geq 41$
- $2x_1 + 5x_2 + 9x_6 \geq 41$
- $5x_2 + 16x_4 + 9x_6 \geq 32$
- $2x_1 + 16x_3 + 9x_6 \geq 32$
- $2x_1 + 5x_2 + 9x_6 \geq 32$
- $5x_2 + 16x_4 + 9x_6 \geq 39$
- $2x_1 + 16x_3 + 9x_6 \geq 39$
- $2x_1 + 5x_2 + 9x_6 \geq 39$
- $2x_1 + 6x_5 \leq 132$
- $16x_3 + 6x_5 \leq 265$
- $2x_1 + 16x_3 \leq 245$
- $2x_1 + 5x_2 \leq 61$
- $5x_2 + 6x_5 \leq 130$
- $16x_3 + 16x_4 \leq 59$
- $16x_4 + 9x_6 \leq 324$
- $16x_4 + 6x_5 \leq 229$
- $16x_3 + 9x_6 \leq 109$
- $2x_1 + 5x_2 + 4x_3 + 16x_4 + 7x_5 + x_6 \leq 109$
- $x_1$ is an integer
- $x_2$ is an integer
- $x_3$ is continuous
- $x_4$ is an integer
- $x_5$ is an integer
- $x_6$ is an integer

## Step 4: Provide the symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Hank'),
        ('x2', 'hours worked by Dale'),
        ('x3', 'hours worked by Laura'),
        ('x4', 'hours worked by George'),
        ('x5', 'hours worked by Peggy'),
        ('x6', 'hours worked by Paul')
    ],
    'objective_function': '8*x1 + 2*x2 + 4*x3 + 4*x4 + 7*x5 + x6',
    'constraints': [
        '2*x1 <= 359',
        '5*x2 <= 359',
        '16*x3 <= 359',
        '16*x4 <= 359',
        '6*x5 <= 359',
        '9*x6 <= 359',
        '5*x2 + 6*x5 >= 31',
        '6*x5 + 9*x6 >= 42',
        '16*x3 + 9*x6 >= 33',
        '16*x3 + 6*x5 >= 33',
        '5*x2 + 16*x4 + 9*x6 >= 41',
        '2*x1 + 16*x3 + 9*x6 >= 41',
        '2*x1 + 5*x2 + 9*x6 >= 41',
        '5*x2 + 16*x4 + 9*x6 >= 32',
        '2*x1 + 16*x3 + 9*x6 >= 32',
        '2*x1 + 5*x2 + 9*x6 >= 32',
        '5*x2 + 16*x4 + 9*x6 >= 39',
        '2*x1 + 16*x3 + 9*x6 >= 39',
        '2*x1 + 5*x2 + 9*x6 >= 39',
        '2*x1 + 6*x5 <= 132',
        '16*x3 + 6*x5 <= 265',
        '2*x1 + 16*x3 <= 245',
        '2*x1 + 5*x2 <= 61',
        '5*x2 + 6*x5 <= 130',
        '16*x3 + 16*x4 <= 59',
        '16*x4 + 9*x6 <= 324',
        '16*x4 + 6*x5 <= 229',
        '16*x3 + 9*x6 <= 109',
        '2*x1 + 5*x2 + 4*x3 + 16*x4 + 7*x5 + x6 <= 109'
    ]
}
```

## Step 5: Write the Gurobi code to solve the optimization problem
```python
import gurobi

def solve_optimization_problem():
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x2 = model.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Dale
    x3 = model.addVar(name='x3')  # hours worked by Laura
    x4 = model.addVar(name='x4', vtype=gurobi.GRB.INTEGER)  # hours worked by George
    x5 = model.addVar(name='x5', vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
    x6 = model.addVar(name='x6', vtype=gurobi.GRB.INTEGER)  # hours worked by Paul

    # Objective function
    model.setObjective(8 * x1 + 2 * x2 + 4 * x3 + 4 * x4 + 7 * x5 + x6, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(2 * x1 <= 359)
    model.addConstr(5 * x2 <= 359)
    model.addConstr(16 * x3 <= 359)
    model.addConstr(16 * x4 <= 359)
    model.addConstr(6 * x5 <= 359)
    model.addConstr(9 * x6 <= 359)

    model.addConstr(5 * x2 + 6 * x5 >= 31)
    model.addConstr(6 * x5 + 9 * x6 >= 42)
    model.addConstr(16 * x3 + 9 * x6 >= 33)
    model.addConstr(16 * x3 + 6 * x5 >= 33)
    model.addConstr(5 * x2 + 16 * x4 + 9 * x6 >= 41)
    model.addConstr(2 * x1 + 16 * x3 + 9 * x6 >= 41)
    model.addConstr(2 * x1 + 5 * x2 + 9 * x6 >= 41)
    model.addConstr(5 * x2 + 16 * x4 + 9 * x6 >= 32)
    model.addConstr(2 * x1 + 16 * x3 + 9 * x6 >= 32)
    model.addConstr(2 * x1 + 5 * x2 + 9 * x6 >= 32)
    model.addConstr(5 * x2 + 16 * x4 + 9 * x6 >= 39)
    model.addConstr(2 * x1 + 16 * x3 + 9 * x6 >= 39)
    model.addConstr(2 * x1 + 5 * x2 + 9 * x6 >= 39)

    model.addConstr(2 * x1 + 6 * x5 <= 132)
    model.addConstr(16 * x3 + 6 * x5 <= 265)
    model.addConstr(2 * x1 + 16 * x3 <= 245)
    model.addConstr(2 * x1 + 5 * x2 <= 61)
    model.addConstr(5 * x2 + 6 * x5 <= 130)
    model.addConstr(16 * x3 + 16 * x4 <= 59)
    model.addConstr(16 * x4 + 9 * x6 <= 324)
    model.addConstr(16 * x4 + 6 * x5 <= 229)
    model.addConstr(16 * x3 + 9 * x6 <= 109)
    model.addConstr(2 * x1 + 5 * x2 + 4 * x3 + 16 * x4 + 7 * x5 + x6 <= 109)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("x1: ", x1.varValue)
        print("x2: ", x2.varValue)
        print("x3: ", x3.varValue)
        print("x4: ", x4.varValue)
        print("x5: ", x5.varValue)
        print("x6: ", x6.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```