## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Jean', 'hours worked by John', 'hours worked by Dale', 'hours worked by Peggy']. Let's denote them as $x_0$, $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Define the symbolic representation of the problem
The objective function to maximize is $5.02x_0 + 3.45x_1 + 6.23x_2 + 5.24x_3$.

## Step 3: List the constraints
1. $2x_0 \geq 2 \cdot x_0$ is not needed as it's a fixed value, similarly for others.
2. $5x_0 \geq 5 \cdot x_0$ is not needed.
3. $3x_1 \geq 3 \cdot x_1$ is not needed.
4. $5x_1 \geq 5 \cdot x_1$ is not needed.
5. $3x_2 \geq 3 \cdot x_2$ is not needed.
6. $1x_2 \geq 1 \cdot x_2$ is not needed.
7. $5x_3 \geq 5 \cdot x_3$ is not needed.
8. $1x_3 \geq 1 \cdot x_3$ is not needed.

## 4: Translate given constraints into algebraic form
- $2x_0 + 3x_1 + 5x_3 \geq 16$
- $2x_0 + 3x_2 + 5x_3 \geq 16$
- $3x_1 + 3x_2 + 5x_3 \geq 16$
- $2x_0 + 3x_1 + 5x_3 \geq 10$
- $2x_0 + 3x_2 + 5x_3 \geq 10$
- $3x_1 + 3x_2 + 5x_3 \geq 10$
- $2x_0 + 3x_1 + 5x_3 \geq 12$
- $2x_0 + 3x_2 + 5x_3 \geq 12$
- $3x_1 + 3x_2 + 5x_3 \geq 12$
- $5x_0 + 1x_3 \geq 7$
- $5x_0 + 5x_1 + 1x_2 \geq 16$
- $3x_1 + 3x_2 \leq 54$
- $2x_0 + 5x_3 \leq 41$
- $2x_0 + 3x_1 + 5x_3 \leq 47$
- $3x_1 + 3x_2 + 5x_3 \leq 43$
- $2x_0 + 3x_2 + 5x_3 \leq 32$
- $2x_0 + 3x_1 + 3x_2 + 5x_3 \leq 32$
- $5x_0 + 1x_3 \leq 54$
- $5x_0 + 5x_1 \leq 53$
- $5x_1 + 1x_2 \leq 18$
- $5x_1 + 1x_2 + 1x_3 \leq 23$
- $5x_0 + 5x_1 + 1x_2 \leq 52$
- $5x_0 + 5x_1 + 1x_2 + 1x_3 \leq 52$

## 5: Create the Gurobi model
We will now create a Gurobi model in Python.

```python
import gurobi

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

# Define the variables
x0 = m.addVar(name='x0', lb=0)  # hours worked by Jean
x1 = m.addVar(name='x1', lb=0)  # hours worked by John
x2 = m.addVar(name='x2', lb=0)  # hours worked by Dale
x3 = m.addVar(name='x3', lb=0)  # hours worked by Peggy

# Define the objective function
m.setObjective(5.02*x0 + 3.45*x1 + 6.23*x2 + 5.24*x3, gurobi.GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x0 + 3*x1 + 5*x3 >= 16)
m.addConstr(2*x0 + 3*x2 + 5*x3 >= 16)
m.addConstr(3*x1 + 3*x2 + 5*x3 >= 16)
m.addConstr(2*x0 + 3*x1 + 5*x3 >= 10)
m.addConstr(2*x0 + 3*x2 + 5*x3 >= 10)
m.addConstr(3*x1 + 3*x2 + 5*x3 >= 10)
m.addConstr(2*x0 + 3*x1 + 5*x3 >= 12)
m.addConstr(2*x0 + 3*x2 + 5*x3 >= 12)
m.addConstr(3*x1 + 3*x2 + 5*x3 >= 12)
m.addConstr(5*x0 + x3 >= 7)
m.addConstr(5*x0 + 5*x1 + x2 >= 16)
m.addConstr(3*x1 + 3*x2 <= 54)
m.addConstr(2*x0 + 5*x3 <= 41)
m.addConstr(2*x0 + 3*x1 + 5*x3 <= 47)
m.addConstr(3*x1 + 3*x2 + 5*x3 <= 43)
m.addConstr(2*x0 + 3*x2 + 5*x3 <= 32)
m.addConstr(2*x0 + 3*x1 + 3*x2 + 5*x3 <= 32)
m.addConstr(5*x0 + x3 <= 54)
m.addConstr(5*x0 + 5*x1 <= 53)
m.addConstr(5*x1 + x2 <= 18)
m.addConstr(5*x1 + x2 + x3 <= 23)
m.addConstr(5*x0 + 5*x1 + x2 <= 52)
m.addConstr(5*x0 + 5*x1 + x2 + x3 <= 52)

# Solve the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print('Objective: ', m.objVal)
    print('Hours worked by Jean: ', x0.varValue)
    print('Hours worked by John: ', x1.varValue)
    print('Hours worked by Dale: ', x2.varValue)
    print('Hours worked by Peggy: ', x3.varValue)
else:
    print('No solution found')
```

## 6: Symbolic representation
```json
{
    'sym_variables': [
        ['x0', 'hours worked by Jean'],
        ['x1', 'hours worked by John'],
        ['x2', 'hours worked by Dale'],
        ['x3', 'hours worked by Peggy']
    ],
    'objective_function': '5.02*x0 + 3.45*x1 + 6.23*x2 + 5.24*x3',
    'constraints': [
        '2*x0 + 3*x1 + 5*x3 >= 16',
        '2*x0 + 3*x2 + 5*x3 >= 16',
        '3*x1 + 3*x2 + 5*x3 >= 16',
        '2*x0 + 3*x1 + 5*x3 >= 10',
        '2*x0 + 3*x2 + 5*x3 >= 10',
        '3*x1 + 3*x2 + 5*x3 >= 10',
        '2*x0 + 3*x1 + 5*x3 >= 12',
        '2*x0 + 3*x2 + 5*x3 >= 12',
        '3*x1 + 3*x2 + 5*x3 >= 12',
        '5*x0 + x3 >= 7',
        '5*x0 + 5*x1 + x2 >= 16',
        '3*x1 + 3*x2 <= 54',
        '2*x0 + 5*x3 <= 41',
        '2*x0 + 3*x1 + 5*x3 <= 47',
        '3*x1 + 3*x2 + 5*x3 <= 43',
        '2*x0 + 3*x2 + 5*x3 <= 32',
        '2*x0 + 3*x1 + 3*x2 + 5*x3 <= 32',
        '5*x0 + x3 <= 54',
        '5*x0 + 5*x1 <= 53',
        '5*x1 + x2 <= 18',
        '5*x1 + x2 + x3 <= 23',
        '5*x0 + 5*x1 + x2 <= 52',
        '5*x0 + 5*x1 + x2 + x3 <= 52'
    ]
}
```