To solve the given optimization problem using Gurobi, we first need to understand and formulate the problem mathematically. The objective is to maximize the value of \(8.96 \times \text{hours worked by Peggy} + 1.78 \times \text{hours worked by Paul} + 9.49 \times \text{hours worked by Bill}\), subject to several constraints involving organization scores.

Let's denote:
- \(x_0\) as the hours worked by Peggy,
- \(x_1\) as the hours worked by Paul, and
- \(x_2\) as the hours worked by Bill.

The organization scores are given as:
- Peggy: 20,
- Paul: 17,
- Bill: 4.

The constraints can be formulated as follows:
1. \(20x_0 + 17x_1 \geq 38\)
2. \(20x_0 + 4x_2 \geq 35\)
3. \(17x_1 + 4x_2 \geq 50\)
4. \(20x_0 + 17x_1 + 4x_2 \geq 83\)
5. \(17x_1 + 4x_2 \leq 198\)
6. \(20x_0 + 17x_1 \leq 166\)
7. \(20x_0 + 17x_1 + 4x_2 \leq 166\)

All variables (\(x_0, x_1, x_2\)) are continuous (can be fractional), which aligns with the problem statement.

Now, let's write the Gurobi code for this optimization problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Peggy")
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Paul")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hours_worked_by_Bill")

# Set the objective function
m.setObjective(8.96*x0 + 1.78*x1 + 9.49*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(20*x0 + 17*x1 >= 38, name="Constraint_1")
m.addConstr(20*x0 + 4*x2 >= 35, name="Constraint_2")
m.addConstr(17*x1 + 4*x2 >= 50, name="Constraint_3")
m.addConstr(20*x0 + 17*x1 + 4*x2 >= 83, name="Constraint_4")
m.addConstr(17*x1 + 4*x2 <= 198, name="Constraint_5")
m.addConstr(20*x0 + 17*x1 <= 166, name="Constraint_6")
m.addConstr(20*x0 + 17*x1 + 4*x2 <= 166, name="Constraint_7")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Peggy: {x0.x}")
    print(f"Hours worked by Paul: {x1.x}")
    print(f"Hours worked by Bill: {x2.x}")
    print(f"Objective value: {m.objVal}")
else:
    print("No optimal solution found")
```