## Step 1: Define the symbolic variables and their corresponding natural language objects.
The variables are 'hours worked by Jean', 'hours worked by Ringo', 'hours worked by John', 'hours worked by Paul', which can be denoted as $x_0$, $x_1$, $x_2$, $x_3$ respectively.

## Step 2: Translate the objective function into an algebraic term using symbolic variables.
The objective function to maximize is $7.17x_0 + 7.08x_1 + 7.84x_2 + 7.77x_3$.

## Step 3: List all the constraints in semi-algebraic form using symbolic variables.
Constraints:
1. $3.56x_0 \leq 173$
2. $9.02x_0 \leq 108$
3. $1.75x_1 \leq 173$
4. $15.5x_1 \leq 108$
5. $11.48x_2 \leq 173$
6. $1.29x_2 \leq 108$
7. $6.79x_3 \leq 173$
8. $6.69x_3 \leq 108$
9. $1.75x_1 + 11.48x_2 + 6.79x_3 \geq 21$
10. $3.56x_0 + 1.75x_1 + 11.48x_2 \geq 21$
11. $3.56x_0 + 11.48x_2 + 6.79x_3 \geq 21$
12. $1.75x_1 + 11.48x_2 + 6.79x_3 \geq 39$
13. $3.56x_0 + 1.75x_1 + 11.48x_2 \geq 39$
14. $3.56x_0 + 11.48x_2 + 6.79x_3 \geq 39$
15. $1.75x_1 + 11.48x_2 + 6.79x_3 \geq 30$
16. $3.56x_0 + 1.75x_1 + 11.48x_2 \geq 30$
17. $3.56x_0 + 11.48x_2 + 6.79x_3 \geq 30$
18. $1.29x_2 + 6.69x_3 \geq 22$
19. $9.02x_0 + 15.5x_1 + 1.29x_2 \geq 24$
20. $9.02x_0 + 15.5x_1 + 6.69x_3 \geq 24$
21. $9.02x_0 + 15.5x_1 + 1.29x_2 \geq 21$
22. $9.02x_0 + 15.5x_1 + 6.69x_3 \geq 21$
23. $3.56x_0 + 6.79x_3 \leq 106$
24. $1.75x_1 + 11.48x_2 + 6.79x_3 \leq 164$
25. $3.56x_0 + 1.75x_1 + 6.79x_3 \leq 157$
26. $3.56x_0 + 1.75x_1 + 11.48x_2 + 6.79x_3 \leq 157$
27. $15.5x_1 + 6.69x_3 \leq 81$
28. $9.02x_0 + 6.69x_3 \leq 87$
29. $15.5x_1 + 1.29x_2 + 6.69x_3 \leq 66$
30. $9.02x_0 + 15.5x_1 + 1.29x_2 \leq 81$
31. $9.02x_0 + 15.5x_1 + 6.69x_3 \leq 50$
32. $9.02x_0 + 15.5x_1 + 1.29x_2 + 6.69x_3 \leq 50$

## Step 4: Provide the symbolic representation of the problem.
```json
{
    'sym_variables': [
        ('x0', 'hours worked by Jean'), 
        ('x1', 'hours worked by Ringo'), 
        ('x2', 'hours worked by John'), 
        ('x3', 'hours worked by Paul')
    ], 
    'objective_function': '7.17*x0 + 7.08*x1 + 7.84*x2 + 7.77*x3', 
    'constraints': [
        '3.56*x0 <= 173',
        '9.02*x0 <= 108',
        '1.75*x1 <= 173',
        '15.5*x1 <= 108',
        '11.48*x2 <= 173',
        '1.29*x2 <= 108',
        '6.79*x3 <= 173',
        '6.69*x3 <= 108',
        '1.75*x1 + 11.48*x2 + 6.79*x3 >= 21',
        '3.56*x0 + 1.75*x1 + 11.48*x2 >= 21',
        '3.56*x0 + 11.48*x2 + 6.79*x3 >= 21',
        '1.75*x1 + 11.48*x2 + 6.79*x3 >= 39',
        '3.56*x0 + 1.75*x1 + 11.48*x2 >= 39',
        '3.56*x0 + 11.48*x2 + 6.79*x3 >= 39',
        '1.75*x1 + 11.48*x2 + 6.79*x3 >= 30',
        '3.56*x0 + 1.75*x1 + 11.48*x2 >= 30',
        '3.56*x0 + 11.48*x2 + 6.79*x3 >= 30',
        '1.29*x2 + 6.69*x3 >= 22',
        '9.02*x0 + 15.5*x1 + 1.29*x2 >= 24',
        '9.02*x0 + 15.5*x1 + 6.69*x3 >= 24',
        '9.02*x0 + 15.5*x1 + 1.29*x2 >= 21',
        '9.02*x0 + 15.5*x1 + 6.69*x3 >= 21',
        '3.56*x0 + 6.79*x3 <= 106',
        '1.75*x1 + 11.48*x2 + 6.79*x3 <= 164',
        '3.56*x0 + 1.75*x1 + 6.79*x3 <= 157',
        '3.56*x0 + 1.75*x1 + 11.48*x2 + 6.79*x3 <= 157',
        '15.5*x1 + 6.69*x3 <= 81',
        '9.02*x0 + 6.69*x3 <= 87',
        '15.5*x1 + 1.29*x2 + 6.69*x3 <= 66',
        '9.02*x0 + 15.5*x1 + 1.29*x2 <= 81',
        '9.02*x0 + 15.5*x1 + 6.69*x3 <= 50',
        '9.02*x0 + 15.5*x1 + 1.29*x2 + 6.69*x3 <= 50'
    ]
}
```

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

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

    # Define variables
    x0 = model.addVar(name="x0", lb=0)  # hours worked by Jean
    x1 = model.addVar(name="x1", lb=0)  # hours worked by Ringo
    x2 = model.addVar(name="x2", lb=0)  # hours worked by John
    x3 = model.addVar(name="x3", lb=0)  # hours worked by Paul

    # Objective function
    model.setObjective(7.17 * x0 + 7.08 * x1 + 7.84 * x2 + 7.77 * x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(3.56 * x0 <= 173)
    model.addConstr(9.02 * x0 <= 108)
    model.addConstr(1.75 * x1 <= 173)
    model.addConstr(15.5 * x1 <= 108)
    model.addConstr(11.48 * x2 <= 173)
    model.addConstr(1.29 * x2 <= 108)
    model.addConstr(6.79 * x3 <= 173)
    model.addConstr(6.69 * x3 <= 108)
    model.addConstr(1.75 * x1 + 11.48 * x2 + 6.79 * x3 >= 21)
    model.addConstr(3.56 * x0 + 1.75 * x1 + 11.48 * x2 >= 21)
    model.addConstr(3.56 * x0 + 11.48 * x2 + 6.79 * x3 >= 21)
    model.addConstr(1.75 * x1 + 11.48 * x2 + 6.79 * x3 >= 39)
    model.addConstr(3.56 * x0 + 1.75 * x1 + 11.48 * x2 >= 39)
    model.addConstr(3.56 * x0 + 11.48 * x2 + 6.79 * x3 >= 39)
    model.addConstr(1.75 * x1 + 11.48 * x2 + 6.79 * x3 >= 30)
    model.addConstr(3.56 * x0 + 1.75 * x1 + 11.48 * x2 >= 30)
    model.addConstr(3.56 * x0 + 11.48 * x2 + 6.79 * x3 >= 30)
    model.addConstr(1.29 * x2 + 6.69 * x3 >= 22)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 1.29 * x2 >= 24)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 6.69 * x3 >= 24)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 1.29 * x2 >= 21)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 6.69 * x3 >= 21)
    model.addConstr(3.56 * x0 + 6.79 * x3 <= 106)
    model.addConstr(1.75 * x1 + 11.48 * x2 + 6.79 * x3 <= 164)
    model.addConstr(3.56 * x0 + 1.75 * x1 + 6.79 * x3 <= 157)
    model.addConstr(3.56 * x0 + 1.75 * x1 + 11.48 * x2 + 6.79 * x3 <= 157)
    model.addConstr(15.5 * x1 + 6.69 * x3 <= 81)
    model.addConstr(9.02 * x0 + 6.69 * x3 <= 87)
    model.addConstr(15.5 * x1 + 1.29 * x2 + 6.69 * x3 <= 66)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 1.29 * x2 <= 81)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 6.69 * x3 <= 50)
    model.addConstr(9.02 * x0 + 15.5 * x1 + 1.29 * x2 + 6.69 * x3 <= 50)

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objVal)
        print("Hours worked by Jean: ", x0.varValue)
        print("Hours worked by Ringo: ", x1.varValue)
        print("Hours worked by John: ", x2.varValue)
        print("Hours worked by Paul: ", x3.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```