## Step 1: Define the symbolic representation of the variables
The variables are 'hours worked by Hank' and 'hours worked by Ringo', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $6.31x_1 + 8.64x_2$.

## Step 3: Define the constraints in symbolic notation
The constraints are:
- $17.41x_1 + 10.0x_2 \geq 49$
- $2.79x_1 + 2.13x_2 \geq 32$
- $5x_1 - x_2 \geq 0$
- $17.41x_1 + 10.0x_2 \leq 161$
- $2.79x_1 + 2.13x_2 \leq 110$
- $x_1$ is an integer
- $x_2$ is a continuous variable

## 4: Provide the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'hours worked by Hank'), ('x2', 'hours worked by Ringo')],
'objective_function': '6.31*x1 + 8.64*x2',
'constraints': [
    '17.41*x1 + 10.0*x2 >= 49',
    '2.79*x1 + 2.13*x2 >= 32',
    '5*x1 - x2 >= 0',
    '17.41*x1 + 10.0*x2 <= 161',
    '2.79*x1 + 2.13*x2 <= 110',
    'x1 is an integer'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="x1", vtype=gurobi.GRB.INTEGER)  # hours worked by Hank
    x2 = model.addVar(name="x2")  # hours worked by Ringo

    # Define the objective function
    model.setObjective(6.31 * x1 + 8.64 * x2, gurobi.GRB.MAXIMIZE)

    # Define the constraints
    model.addConstr(17.41 * x1 + 10.0 * x2 >= 49)
    model.addConstr(2.79 * x1 + 2.13 * x2 >= 32)
    model.addConstr(5 * x1 - x2 >= 0)
    model.addConstr(17.41 * x1 + 10.0 * x2 <= 161)
    model.addConstr(2.79 * x1 + 2.13 * x2 <= 110)

    # Set bounds for x1 and x2
    model.addConstr(x1 >= 0)
    model.addConstr(x2 >= 0)

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Hours worked by Hank: {x1.varValue}")
        print(f"Hours worked by Ringo: {x2.varValue}")
        print(f"Objective function value: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```