## Step 1: Define the optimization problem and its components
The problem requires maximizing an objective function with three variables: hours worked by Dale ($x_0$), hours worked by John ($x_1$), and hours worked by George ($x_2$). The objective function to maximize is $2.25x_0^2 + 5.18x_0x_1 + 4.6x_0x_2 + 2.19x_1x_2 + 8.59x_0 + 9.2x_1 + 10.0x_2$.

## 2: List all given constraints
1. $x_0 \geq 0$, $x_1 \geq 0$, $x_2 \geq 0$ (Implicit non-negativity constraints)
2. The likelihood to quit index for Dale is 5, for John is 7, and for George is 20.
3. The dollar cost per hour for Dale, John, and George is 16 each.
4. $5x_0^2 + 20x_2^2 \geq 29$
5. $5x_0 + 7x_1 + 20x_2 \geq 32$
6. $7x_1 + 20x_2 \leq 71$
7. $5x_0 + 7x_1 \leq 72$
8. $5x_0 + 7x_1 + 20x_2 \leq 175$
9. $16x_0 + 16x_1 \leq 179$
10. $16x_0 + 16x_2 \leq 136$
11. $16x_0 + 16x_1 + 16x_2 \leq 200$

## 3: Convert the problem into Gurobi code
To solve this problem using Gurobi, we first need to import the Gurobi library and then define the model, variables, objective function, and constraints.

```python
import gurobi as gp

# Define the model
model = gp.Model("optimization_problem")

# Define the variables
x0 = model.addVar(name="hours_worked_by_Dale", lb=0)  # hours worked by Dale
x1 = model.addVar(name="hours_worked_by_John", lb=0)   # hours worked by John
x2 = model.addVar(name="hours_worked_by_George", lb=0) # hours worked by George

# Define the objective function
model.setObjective(2.25*x0**2 + 5.18*x0*x1 + 4.6*x0*x2 + 2.19*x1*x2 + 8.59*x0 + 9.2*x1 + 10.0*x2, gp.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(5*x0**2 + 20*x2**2 >= 29, name="likelihood_quit_index_constraint_1")
model.addConstr(5*x0 + 7*x1 + 20*x2 >= 32, name="likelihood_quit_index_constraint_2")
model.addConstr(7*x1 + 20*x2 <= 71, name="likelihood_quit_index_constraint_3")
model.addConstr(5*x0 + 7*x1 <= 72, name="likelihood_quit_index_constraint_4")
model.addConstr(5*x0 + 7*x1 + 20*x2 <= 175, name="likelihood_quit_index_constraint_5")
model.addConstr(16*x0 + 16*x1 <= 179, name="dollar_cost_constraint_1")
model.addConstr(16*x0 + 16*x2 <= 136, name="dollar_cost_constraint_2")
model.addConstr(16*x0 + 16*x1 + 16*x2 <= 200, name="dollar_cost_constraint_3")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Hours worked by Dale: {x0.varValue}")
    print(f"Hours worked by John: {x1.varValue}")
    print(f"Hours worked by George: {x2.varValue}")
    print(f"Objective function value: {model.objVal}")
else:
    print("No optimal solution found.")
```