## 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 Mary'
- 'x3' corresponds to 'hours worked by Peggy'
- 'x4' corresponds to 'hours worked by Hank'
- 'x5' corresponds to 'hours worked by Bill'

## Step 2: Define the coefficients for the objective function
The objective function to maximize is: $4x_1 + 8x_2 + 6x_3 + 9x_4 + 7x_5$

## Step 3: Define the constraints
The constraints are:
- $1.04x_2 + 5.57x_4 \geq 41$
- $1.04x_2 + 5.86x_3 \geq 21$
- $2.32x_1 + 1.04x_2 \geq 29$
- $5.86x_3 + 2.05x_5 \geq 39$
- $5.57x_4 + 2.05x_5 \geq 17$
- $1.04x_2 + 5.57x_4 \leq 206$
- $2.32x_1 + 2.05x_5 \leq 99$
- $2.32x_1 + 1.04x_2 \leq 193$
- $2.32x_1 + 5.86x_3 \leq 148$
- $1.04x_2 + 5.86x_3 + 5.57x_4 \leq 107$
- $2.32x_1 + 1.04x_2 + 5.86x_3 + 5.57x_4 + 2.05x_5 \leq 224$
- $x_1 \in \mathbb{Z}$
- $x_2 \in \mathbb{Z}$
- $x_3 \in \mathbb{Z}$
- $x_4 \in \mathbb{Z}$
- $x_5 \in \mathbb{Z}$

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

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

# Define the variables
x1 = model.addVar(name='x1', vtype='I')  # hours worked by Dale
x2 = model.addVar(name='x2', vtype='I')  # hours worked by Mary
x3 = model.addVar(name='x3', vtype='I')  # hours worked by Peggy
x4 = model.addVar(name='x4', vtype='I')  # hours worked by Hank
x5 = model.addVar(name='x5', vtype='I')  # hours worked by Bill

# Define the objective function
model.setObjective(4 * x1 + 8 * x2 + 6 * x3 + 9 * x4 + 7 * x5, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(1.04 * x2 + 5.57 * x4 >= 41)
model.addConstr(1.04 * x2 + 5.86 * x3 >= 21)
model.addConstr(2.32 * x1 + 1.04 * x2 >= 29)
model.addConstr(5.86 * x3 + 2.05 * x5 >= 39)
model.addConstr(5.57 * x4 + 2.05 * x5 >= 17)
model.addConstr(1.04 * x2 + 5.57 * x4 <= 206)
model.addConstr(2.32 * x1 + 2.05 * x5 <= 99)
model.addConstr(2.32 * x1 + 1.04 * x2 <= 193)
model.addConstr(2.32 * x1 + 5.86 * x3 <= 148)
model.addConstr(1.04 * x2 + 5.86 * x3 + 5.57 * x4 <= 107)
model.addConstr(2.32 * x1 + 1.04 * x2 + 5.86 * x3 + 5.57 * x4 + 2.05 * x5 <= 224)

# Update model
model.update()

# 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)
else:
    print('The model is infeasible')
```

## 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Dale'),
        ('x2', 'hours worked by Mary'),
        ('x3', 'hours worked by Peggy'),
        ('x4', 'hours worked by Hank'),
        ('x5', 'hours worked by Bill')
    ],
    'objective_function': '4*x1 + 8*x2 + 6*x3 + 9*x4 + 7*x5',
    'constraints': [
        '1.04*x2 + 5.57*x4 >= 41',
        '1.04*x2 + 5.86*x3 >= 21',
        '2.32*x1 + 1.04*x2 >= 29',
        '5.86*x3 + 2.05*x5 >= 39',
        '5.57*x4 + 2.05*x5 >= 17',
        '1.04*x2 + 5.57*x4 <= 206',
        '2.32*x1 + 2.05*x5 <= 99',
        '2.32*x1 + 1.04*x2 <= 193',
        '2.32*x1 + 5.86*x3 <= 148',
        '1.04*x2 + 5.86*x3 + 5.57*x4 <= 107',
        '2.32*x1 + 1.04*x2 + 5.86*x3 + 5.57*x4 + 2.05*x5 <= 224',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ',
        'x3 ∈ ℤ',
        'x4 ∈ ℤ',
        'x5 ∈ ℤ'
    ]
}
```