To solve the given optimization problem using Gurobi, we first need to translate the natural language description into a mathematical formulation. The objective is to maximize the function $6x_0 + 7x_1 + 2x_2$, where $x_0$, $x_1$, and $x_2$ represent the hours worked by Hank, Mary, and Bobby, respectively.

The constraints given are:
1. The total combined dollar cost per hour from all workers must be at least 51.
2. A specific linear combination of hours worked by Hank and Bobby must be non-negative ($2x_0 - 3x_2 \geq 0$).
3. The total combined dollar cost per hour from Hank and Bobby must not exceed 114.
4. The total combined dollar cost per hour from Mary and Bobby must not exceed 109.
5. The total combined dollar cost per hour from all workers must not exceed 109.

Let's denote the hours worked by Hank, Mary, and Bobby as $x_0$, $x_1$, and $x_2$, respectively. Given the dollar costs per hour for each worker ($23.13$ for Hank, $19.71$ for Mary, and $7.57$ for Bobby), we can formulate these constraints mathematically:

- The objective function to maximize: $6x_0 + 7x_1 + 2x_2$
- Constraints:
  - Total cost constraint: $23.13x_0 + 19.71x_1 + 7.57x_2 \geq 51$
  - Linear combination of hours worked by Hank and Bobby: $2x_0 - 3x_2 \geq 0$
  - Cost constraint for Hank and Bobby: $23.13x_0 + 7.57x_2 \leq 114$
  - Cost constraint for Mary and Bobby: $19.71x_1 + 7.57x_2 \leq 109$
  - Total cost constraint for all workers (note: this is less than or equal to 109, not more): $23.13x_0 + 19.71x_1 + 7.57x_2 \leq 109$

All variables ($x_0$, $x_1$, $x_2$) are non-negative since they represent hours worked.

Here is the Gurobi code to solve this optimization problem:
```python
from gurobipy import *

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

# Define variables
x0 = m.addVar(lb=0, name="hours_worked_by_Hank")
x1 = m.addVar(lb=0, name="hours_worked_by_Mary")
x2 = m.addVar(lb=0, name="hours_worked_by_Bobby")

# Set the objective function
m.setObjective(6*x0 + 7*x1 + 2*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(23.13*x0 + 19.71*x1 + 7.57*x2 >= 51, name="total_cost_constraint")
m.addConstr(2*x0 - 3*x2 >= 0, name="hank_bobby_hours_constraint")
m.addConstr(23.13*x0 + 7.57*x2 <= 114, name="hank_bobby_cost_constraint")
m.addConstr(19.71*x1 + 7.57*x2 <= 109, name="mary_bobby_cost_constraint")
m.addConstr(23.13*x0 + 19.71*x1 + 7.57*x2 <= 109, name="total_cost_constraint_all")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hours worked by Hank: {x0.x}")
    print(f"Hours worked by Mary: {x1.x}")
    print(f"Hours worked by Bobby: {x2.x}")
else:
    print("No optimal solution found")
```