To solve this optimization problem, we first need to define the variables, objective function, and constraints using Gurobi's Python API.

### Problem Definition

We have five variables:
- \(H_H\): hours worked by Hank
- \(H_B\): hours worked by Bobby
- \(H_L\): hours worked by Laura
- \(H_P\): hours worked by Paul
- \(H_{Pe}\): hours worked by Peggy

### Objective Function

Maximize: \(2.55H_H + 7.28H_B + 5.19H_L + 8.05H_P + 9.52H_{Pe}\)

### Constraints

1. \(30H_H \leq 482\)
2. \(3H_H \leq 202\)
3. \(14H_B \leq 482\)
4. \(18H_B \leq 202\)
5. \(25H_L \leq 482\)
6. \(20H_L \leq 202\)
7. \(8H_P \leq 482\)
8. \(24H_P \leq 202\)
9. \(14H_{Pe} \leq 482\)
10. \(9H_{Pe} \leq 202\)
11. \(25H_L + 8H_P \geq 59\)
12. \(30H_H + 25H_L \geq 92\)
13. \(14H_B + 25H_L \geq 57\)
14. \(14H_B + 14H_{Pe} \geq 54\)
15. \(14H_B + 8H_P \geq 74\)
16. \(30H_H + 14H_B + 8H_P \geq 58\)
17. \(14H_B + 25H_L + 14H_{Pe} \geq 58\)
18. \(30H_H + 14H_B + 8H_P \geq 51\)
19. \(14H_B + 25H_L + 14H_{Pe} \geq 51\)
20. \(20H_L + 24H_P + 9H_{Pe} \geq 25\)
21. \(18H_B + 20H_L + 24H_P \geq 25\)
22. \(20H_L + 24H_P + 9H_{Pe} \geq 29\)
23. \(18H_B + 20H_L + 24H_P \geq 29\)
24. \(30H_H + 14H_{Pe} \leq 118\)
25. \(30H_H + 25H_L \leq 230\)
26. \(14H_B + 14H_{Pe} \leq 111\)
27. \(25H_L + 14H_{Pe} \leq 201\)
28. \(30H_H + 8H_P \leq 444\)
29. \(30H_H + 14H_B + 25H_L + 8H_P + 14H_{Pe} \leq 444\)
30. \(24H_P + 9H_{Pe} \leq 128\)
31. \(3H_H + 9H_{Pe} \leq 76\)
32. \(3H_H + 24H_P \leq 110\)
33. \(3H_H + 18H_B \leq 47\)
34. \(3H_H + 20H_L \leq 137\)
35. \(3H_H + 18H_B + 20H_L \leq 97\)
36. \(18H_B + 20H_L + 9H_{Pe} \leq 162\)
37. \(3H_H + 24H_P + 9H_{Pe} \leq 47\)
38. \(3H_H + 18H_B + 24H_P \leq 201\)
39. \(3H_H + 20H_L + 9H_{Pe} \leq 181\)
40. \(18H_B + 20H_L + 24H_P \leq 103\)
41. \(3H_H + 18H_B + 20H_L + 24H_P + 9H_{Pe} \leq 103\)

### Gurobi Code

```python
import gurobi

def solve_optimization_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define variables
    HH = model.addVar(name="HH", vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    HB = model.addVar(name="HB", vtype=gurobi.GRB.INTEGER)  # hours worked by Bobby
    HL = model.addVar(name="HL", vtype=gurobi.GRB.INTEGER)  # hours worked by Laura
    HP = model.addVar(name="HP", vtype=gurobi.GRB.INTEGER)  # hours worked by Paul
    HPe = model.addVar(name="HPe")  # hours worked by Peggy

    # Objective function
    model.setObjective(2.55 * HH + 7.28 * HB + 5.19 * HL + 8.05 * HP + 9.52 * HPe, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(30 * HH <= 482)
    model.addConstr(3 * HH <= 202)
    model.addConstr(14 * HB <= 482)
    model.addConstr(18 * HB <= 202)
    model.addConstr(25 * HL <= 482)
    model.addConstr(20 * HL <= 202)
    model.addConstr(8 * HP <= 482)
    model.addConstr(24 * HP <= 202)
    model.addConstr(14 * HPe <= 482)
    model.addConstr(9 * HPe <= 202)

    model.addConstr(25 * HL + 8 * HP >= 59)
    model.addConstr(30 * HH + 25 * HL >= 92)
    model.addConstr(14 * HB + 25 * HL >= 57)
    model.addConstr(14 * HB + 14 * HPe >= 54)
    model.addConstr(14 * HB + 8 * HP >= 74)
    model.addConstr(30 * HH + 14 * HB + 8 * HP >= 58)
    model.addConstr(14 * HB + 25 * HL + 14 * HPe >= 58)
    model.addConstr(30 * HH + 14 * HB + 8 * HP >= 51)
    model.addConstr(14 * HB + 25 * HL + 14 * HPe >= 51)

    model.addConstr(20 * HL + 24 * HP + 9 * HPe >= 25)
    model.addConstr(18 * HB + 20 * HL + 24 * HP >= 25)
    model.addConstr(20 * HL + 24 * HP + 9 * HPe >= 29)
    model.addConstr(18 * HB + 20 * HL + 24 * HP >= 29)

    model.addConstr(30 * HH + 14 * HPe <= 118)
    model.addConstr(30 * HH + 25 * HL <= 230)
    model.addConstr(14 * HB + 14 * HPe <= 111)
    model.addConstr(25 * HL + 14 * HPe <= 201)
    model.addConstr(30 * HH + 8 * HP <= 444)
    model.addConstr(30 * HH + 14 * HB + 25 * HL + 8 * HP + 14 * HPe <= 444)

    model.addConstr(24 * HP + 9 * HPe <= 128)
    model.addConstr(3 * HH + 9 * HPe <= 76)
    model.addConstr(3 * HH + 24 * HP <= 110)
    model.addConstr(3 * HH + 18 * HB <= 47)
    model.addConstr(3 * HH + 20 * HL <= 137)
    model.addConstr(3 * HH + 18 * HB + 20 * HL <= 97)
    model.addConstr(18 * HB + 20 * HL + 9 * HPe <= 162)
    model.addConstr(3 * HH + 24 * HP + 9 * HPe <= 47)
    model.addConstr(3 * HH + 18 * HB + 24 * HP <= 201)
    model.addConstr(3 * HH + 20 * HL + 9 * HPe <= 181)
    model.addConstr(18 * HB + 20 * HL + 24 * HP <= 103)
    model.addConstr(3 * HH + 18 * HB + 20 * HL + 24 * HP + 9 * HPe <= 103)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Objective: ", model.objval)
        print("HH: ", HH.varValue)
        print("HB: ", HB.varValue)
        print("HL: ", HL.varValue)
        print("HP: ", HP.varValue)
        print("HPe: ", HPe.varValue)
    else:
        print("The model is infeasible")

solve_optimization_problem()
```