## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Hank', 'hours worked by Bobby', 'hours worked by Laura', 'hours worked by Paul', 'hours worked by Peggy'] which can be represented symbolically as 'x0', 'x1', 'x2', 'x3', 'x4' respectively.

## Step 2: Define the symbolic representation of the problem
The symbolic representation of the variables and their corresponding natural language objects are:
- 'x0' : 'hours worked by Hank'
- 'x1' : 'hours worked by Bobby'
- 'x2' : 'hours worked by Laura'
- 'x3' : 'hours worked by Paul'
- 'x4' : 'hours worked by Peggy'

## Step 3: Formulate the objective function
The objective function to maximize is: $2.55x_0 + 7.28x_1 + 5.19x_2 + 8.05x_3 + 9.52x_4$

## 4: List the constraints
The constraints are:
- $30x_0 \leq 482$
- $3x_0 \leq 202$
- $14x_1 \leq 482$
- $18x_1 \leq 202$
- $25x_2 \leq 482$
- $20x_2 \leq 202$
- $8x_3 \leq 482$
- $24x_3 \leq 202$
- $14x_4 \leq 482$
- $9x_4 \leq 202$
- $25x_2 + 8x_3 \geq 59$
- $30x_0 + 25x_2 \geq 92$
- $14x_1 + 25x_2 \geq 57$
- $14x_1 + 14x_4 \geq 54$
- $14x_1 + 8x_3 \geq 74$
- $30x_0 + 14x_1 + 8x_3 \geq 58$
- $14x_1 + 25x_2 + 14x_4 \geq 58$
- $30x_0 + 14x_1 + 8x_3 \geq 51$
- $14x_1 + 25x_2 + 14x_4 \geq 51$
- $20x_2 + 24x_3 + 9x_4 \geq 25$
- $18x_1 + 20x_2 + 24x_3 \geq 25$
- $20x_2 + 24x_3 + 9x_4 \geq 29$
- $18x_1 + 20x_2 + 24x_3 \geq 29$
- $30x_0 + 14x_4 \leq 118$
- $30x_0 + 25x_2 \leq 230$
- $14x_1 + 14x_4 \leq 111$
- $25x_2 + 14x_4 \leq 201$
- $30x_0 + 8x_3 \leq 444$
- $30x_0 + 14x_1 + 25x_2 + 8x_3 + 14x_4 \leq 444$
- $24x_3 + 9x_4 \leq 128$
- $3x_0 + 9x_4 \leq 76$
- $3x_0 + 24x_3 \leq 110$
- $3x_0 + 18x_1 \leq 47$
- $3x_0 + 20x_2 \leq 137$
- $3x_0 + 18x_1 + 20x_2 \leq 97$
- $18x_1 + 20x_2 + 9x_4 \leq 162$
- $3x_0 + 24x_3 + 9x_4 \leq 47$
- $3x_0 + 18x_1 + 24x_3 \leq 201$
- $3x_0 + 20x_2 + 9x_4 \leq 181$
- $18x_1 + 20x_2 + 24x_3 \leq 103$
- $3x_0 + 18x_1 + 20x_2 + 24x_3 + 9x_4 \leq 103$
- $x_0 \in \mathbb{Z}$
- $x_1 \in \mathbb{Z}$
- $x_2 \in \mathbb{Z}$
- $x_3 \in \mathbb{Z}$

## 5: Write the Gurobi code
```python
import gurobi

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

# Define the variables
x0 = m.addVar(name='x0', vtype='I')  # hours worked by Hank
x1 = m.addVar(name='x1', vtype='I')  # hours worked by Bobby
x2 = m.addVar(name='x2', vtype='I')  # hours worked by Laura
x3 = m.addVar(name='x3', vtype='I')  # hours worked by Paul
x4 = m.addVar(name='x4')  # hours worked by Peggy

# Objective function
m.setObjective(2.55*x0 + 7.28*x1 + 5.19*x2 + 8.05*x3 + 9.52*x4, gurobi.GRB.MAXIMIZE)

# Constraints
m.addConstr(30*x0 <= 482)
m.addConstr(3*x0 <= 202)
m.addConstr(14*x1 <= 482)
m.addConstr(18*x1 <= 202)
m.addConstr(25*x2 <= 482)
m.addConstr(20*x2 <= 202)
m.addConstr(8*x3 <= 482)
m.addConstr(24*x3 <= 202)
m.addConstr(14*x4 <= 482)
m.addConstr(9*x4 <= 202)
m.addConstr(25*x2 + 8*x3 >= 59)
m.addConstr(30*x0 + 25*x2 >= 92)
m.addConstr(14*x1 + 25*x2 >= 57)
m.addConstr(14*x1 + 14*x4 >= 54)
m.addConstr(14*x1 + 8*x3 >= 74)
m.addConstr(30*x0 + 14*x1 + 8*x3 >= 58)
m.addConstr(14*x1 + 25*x2 + 14*x4 >= 58)
m.addConstr(30*x0 + 14*x1 + 8*x3 >= 51)
m.addConstr(14*x1 + 25*x2 + 14*x4 >= 51)
m.addConstr(20*x2 + 24*x3 + 9*x4 >= 25)
m.addConstr(18*x1 + 20*x2 + 24*x3 >= 25)
m.addConstr(20*x2 + 24*x3 + 9*x4 >= 29)
m.addConstr(18*x1 + 20*x2 + 24*x3 >= 29)
m.addConstr(30*x0 + 14*x4 <= 118)
m.addConstr(30*x0 + 25*x2 <= 230)
m.addConstr(14*x1 + 14*x4 <= 111)
m.addConstr(25*x2 + 14*x4 <= 201)
m.addConstr(30*x0 + 8*x3 <= 444)
m.addConstr(30*x0 + 14*x1 + 25*x2 + 8*x3 + 14*x4 <= 444)
m.addConstr(24*x3 + 9*x4 <= 128)
m.addConstr(3*x0 + 9*x4 <= 76)
m.addConstr(3*x0 + 24*x3 <= 110)
m.addConstr(3*x0 + 18*x1 <= 47)
m.addConstr(3*x0 + 20*x2 <= 137)
m.addConstr(3*x0 + 18*x1 + 20*x2 <= 97)
m.addConstr(18*x1 + 20*x2 + 9*x4 <= 162)
m.addConstr(3*x0 + 24*x3 + 9*x4 <= 47)
m.addConstr(3*x0 + 18*x1 + 24*x3 <= 201)
m.addConstr(3*x0 + 20*x2 + 9*x4 <= 181)
m.addConstr(18*x1 + 20*x2 + 24*x3 <= 103)
m.addConstr(3*x0 + 18*x1 + 20*x2 + 24*x3 + 9*x4 <= 103)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objval)
    print('x0: ', x0.varValue)
    print('x1: ', x1.varValue)
    print('x2: ', x2.varValue)
    print('x3: ', x3.varValue)
    print('x4: ', x4.varValue)
else:
    print('No optimal solution found')
```

## 6: Symbolic representation of the problem
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Hank'),
        ('x1', 'hours worked by Bobby'),
        ('x2', 'hours worked by Laura'),
        ('x3', 'hours worked by Paul'),
        ('x4', 'hours worked by Peggy')
    ],
    'objective_function': '2.55*x0 + 7.28*x1 + 5.19*x2 + 8.05*x3 + 9.52*x4',
    'constraints': [
        '30*x0 <= 482',
        '3*x0 <= 202',
        '14*x1 <= 482',
        '18*x1 <= 202',
        '25*x2 <= 482',
        '20*x2 <= 202',
        '8*x3 <= 482',
        '24*x3 <= 202',
        '14*x4 <= 482',
        '9*x4 <= 202',
        '25*x2 + 8*x3 >= 59',
        '30*x0 + 25*x2 >= 92',
        '14*x1 + 25*x2 >= 57',
        '14*x1 + 14*x4 >= 54',
        '14*x1 + 8*x3 >= 74',
        '30*x0 + 14*x1 + 8*x3 >= 58',
        '14*x1 + 25*x2 + 14*x4 >= 58',
        '30*x0 + 14*x1 + 8*x3 >= 51',
        '14*x1 + 25*x2 + 14*x4 >= 51',
        '20*x2 + 24*x3 + 9*x4 >= 25',
        '18*x1 + 20*x2 + 24*x3 >= 25',
        '20*x2 + 24*x3 + 9*x4 >= 29',
        '18*x1 + 20*x2 + 24*x3 >= 29',
        '30*x0 + 14*x4 <= 118',
        '30*x0 + 25*x2 <= 230',
        '14*x1 + 14*x4 <= 111',
        '25*x2 + 14*x4 <= 201',
        '30*x0 + 8*x3 <= 444',
        '30*x0 + 14*x1 + 25*x2 + 8*x3 + 14*x4 <= 444',
        '24*x3 + 9*x4 <= 128',
        '3*x0 + 9*x4 <= 76',
        '3*x0 + 24*x3 <= 110',
        '3*x0 + 18*x1 <= 47',
        '3*x0 + 20*x2 <= 137',
        '3*x0 + 18*x1 + 20*x2 <= 97',
        '18*x1 + 20*x2 + 9*x4 <= 162',
        '3*x0 + 24*x3 + 9*x4 <= 47',
        '3*x0 + 18*x1 + 24*x3 <= 201',
        '3*x0 + 20*x2 + 9*x4 <= 181',
        '18*x1 + 20*x2 + 24*x3 <= 103',
        '3*x0 + 18*x1 + 20*x2 + 24*x3 + 9*x4 <= 103'
    ]
}
```