## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank', 'hours worked by Mary', and 'hours worked by Bobby', which can be represented symbolically as $x_1$, $x_2$, and $x_3$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The symbolic representation of the variables is:
- $x_1$ : hours worked by Hank
- $x_2$ : hours worked by Mary
- $x_3$ : hours worked by Bobby

## 3: Define the objective function in symbolic notation
The objective function to maximize is $6x_1 + 7x_2 + 2x_3$.

## 4: Define the constraints in symbolic notation
The constraints are:
1. $23.13x_1 + 19.71x_2 + 7.57x_3 \geq 51$
2. $2x_1 - 3x_3 \geq 0$
3. $23.13x_1 + 7.57x_3 \leq 114$
4. $19.71x_2 + 7.57x_3 \leq 109$
5. $23.13x_1 + 19.71x_2 + 7.57x_3 \leq 109$

## 5: Provide the symbolic representation in the required format
```json
{
'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Mary'), ('x3', 'hours worked by Bobby')],
'objective_function': '6*x1 + 7*x2 + 2*x3',
'constraints': [
'23.13*x1 + 19.71*x2 + 7.57*x3 >= 51',
'2*x1 - 3*x3 >= 0',
'23.13*x1 + 7.57*x3 <= 114',
'19.71*x2 + 7.57*x3 <= 109',
'23.13*x1 + 19.71*x2 + 7.57*x3 <= 109'
]
}
```

## Step 6: Write the Gurobi code to solve the optimization problem
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name='hours_worked_by_Hank', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x2 = model.addVar(name='hours_worked_by_Mary', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
    x3 = model.addVar(name='hours_worked_by_Bobby', lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

    # Define the objective function
    model.setObjective(6 * x1 + 7 * x2 + 2 * x3, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(23.13 * x1 + 19.71 * x2 + 7.57 * x3 >= 51)
    model.addConstr(2 * x1 - 3 * x3 >= 0)
    model.addConstr(23.13 * x1 + 7.57 * x3 <= 114)
    model.addConstr(19.71 * x2 + 7.57 * x3 <= 109)
    model.addConstr(23.13 * x1 + 19.71 * x2 + 7.57 * x3 <= 109)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Hours worked by Hank: ', x1.x)
        print('Hours worked by Mary: ', x2.x)
        print('Hours worked by Bobby: ', x3.x)
    else:
        print('The model is infeasible')

solve_optimization_problem()
```