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

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is: 
\[ 7.48x_1 + 6.82x_2 + 4.25x_3 + 2.38x_4 + 7.91x_5 \]

## 3: List the constraints in symbolic notation
The constraints are:
- \( 14x_2 + 5x_4 \geq 33 \)
- \( 11x_3 + 12x_5 \geq 31 \)
- \( 2x_1 + 14x_2 \geq 44 \)
- \( 14x_2 + 11x_3 \geq 24 \)
- \( 14x_2 + 12x_5 \geq 33 \)
- \( 5x_4 + 12x_5 \geq 20 \)
- \( 2x_1 + 12x_5 \geq 38 \)
- \( 11x_3 + 5x_4 \geq 22 \)
- \( 14x_2 + 5x_4 \leq 231 \)
- \( 14x_2 + 11x_3 \leq 178 \)
- \( 2x_1 + 11x_3 \leq 232 \)
- \( 14x_2 + 11x_3 + 12x_5 \leq 133 \)
- \( 2x_1 + 14x_2 + 11x_3 + 5x_4 + 12x_5 \leq 133 \)
- \( x_1 \in \mathbb{Z} \)
- \( x_2 \in \mathbb{Z} \)
- \( x_3 \in \mathbb{R} \)
- \( x_4 \in \mathbb{R} \)
- \( x_5 \in \mathbb{R} \)

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

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

# Define the variables
x1 = m.addVar(name='x1', vtype=gurobi.GRB.INTEGER)  # hours worked by Peggy
x2 = m.addVar(name='x2', vtype=gurobi.GRB.INTEGER)  # hours worked by Dale
x3 = m.addVar(name='x3')  # hours worked by Bill
x4 = m.addVar(name='x4')  # hours worked by Mary
x5 = m.addVar(name='x5')  # hours worked by Bobby

# Define the objective function
m.setObjective(7.48 * x1 + 6.82 * x2 + 4.25 * x3 + 2.38 * x4 + 7.91 * x5, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(14 * x2 + 5 * x4 >= 33)
m.addConstr(11 * x3 + 12 * x5 >= 31)
m.addConstr(2 * x1 + 14 * x2 >= 44)
m.addConstr(14 * x2 + 11 * x3 >= 24)
m.addConstr(14 * x2 + 12 * x5 >= 33)
m.addConstr(5 * x4 + 12 * x5 >= 20)
m.addConstr(2 * x1 + 12 * x5 >= 38)
m.addConstr(11 * x3 + 5 * x4 >= 22)
m.addConstr(14 * x2 + 5 * x4 <= 231)
m.addConstr(14 * x2 + 11 * x3 <= 178)
m.addConstr(2 * x1 + 11 * x3 <= 232)
m.addConstr(14 * x2 + 11 * x3 + 12 * x5 <= 133)
m.addConstr(2 * x1 + 14 * x2 + 11 * x3 + 5 * x4 + 12 * x5 <= 133)

# Set bounds for variables (not necessary in this case but can be useful)
m.addConstr(x1 >= 0)
m.addConstr(x2 >= 0)
m.addConstr(x3 >= 0)
m.addConstr(x4 >= 0)
m.addConstr(x5 >= 0)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by Peggy: ', x1.varValue)
    print('Hours worked by Dale: ', x2.varValue)
    print('Hours worked by Bill: ', x3.varValue)
    print('Hours worked by Mary: ', x4.varValue)
    print('Hours worked by Bobby: ', x5.varValue)
else:
    print('The model is infeasible')
```

## Step 5: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x1', 'hours worked by Peggy'),
        ('x2', 'hours worked by Dale'),
        ('x3', 'hours worked by Bill'),
        ('x4', 'hours worked by Mary'),
        ('x5', 'hours worked by Bobby')
    ],
    'objective_function': '7.48*x1 + 6.82*x2 + 4.25*x3 + 2.38*x4 + 7.91*x5',
    'constraints': [
        '14*x2 + 5*x4 >= 33',
        '11*x3 + 12*x5 >= 31',
        '2*x1 + 14*x2 >= 44',
        '14*x2 + 11*x3 >= 24',
        '14*x2 + 12*x5 >= 33',
        '5*x4 + 12*x5 >= 20',
        '2*x1 + 12*x5 >= 38',
        '11*x3 + 5*x4 >= 22',
        '14*x2 + 5*x4 <= 231',
        '14*x2 + 11*x3 <= 178',
        '2*x1 + 11*x3 <= 232',
        '14*x2 + 11*x3 + 12*x5 <= 133',
        '2*x1 + 14*x2 + 11*x3 + 5*x4 + 12*x5 <= 133'
    ]
}
```