## Step 1: Define the symbolic variables
Let's define the symbolic variables to represent the natural language objects in the problem description:
- 'x1' for the number of new cooks
- 'x2' for the number of senior cooks

## Step 2: Formulate the objective function
The objective is to minimize the weekly wage bill. The wage for new cooks is $500 a week, and for senior cooks, it is $1000 a week. Therefore, the objective function can be represented as:
\[ \text{Minimize:} \quad 500x_1 + 1000x_2 \]

## 3: Define the constraints
Based on the problem description, we have the following constraints:
1. The weekly wage bill must be kept below $50,000:
\[ 500x_1 + 1000x_2 \leq 50000 \]
2. There must be a minimum of 30 total cooks:
\[ x_1 + x_2 \geq 30 \]
3. At least 5 cooks must be senior cooks:
\[ x_2 \geq 5 \]
4. The number of senior cooks should be at least a third of the number of new cooks:
\[ x_2 \geq \frac{1}{3}x_1 \]
Also, the number of cooks cannot be negative:
\[ x_1 \geq 0, x_2 \geq 0 \]

## 4: Convert to Gurobi code
Now, let's convert this into Gurobi code in Python:

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="new_cooks")
x2 = model.addVar(lb=0, name="senior_cooks")

# Objective function: minimize the wage bill
model.setObjective(500*x1 + 1000*x2, gurobi.GRB.MINIMIZE)

# Constraints
model.addConstr(500*x1 + 1000*x2 <= 50000, name="wage_bill")
model.addConstr(x1 + x2 >= 30, name="total_cooks")
model.addConstr(x2 >= 5, name="senior_cooks_min")
model.addConstr(x2 >= (1/3)*x1, name="senior_to_new_ratio")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of new cooks: {x1.varValue}")
    print(f"Number of senior cooks: {x2.varValue}")
    print(f"Minimum wage bill: ${model.objVal:.2f}")
else:
    print("No optimal solution found.")
```

## 5: Symbolic Representation
The symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'new cooks'), ('x2', 'senior cooks')],
    'objective_function': '500*x1 + 1000*x2',
    'constraints': [
        '500*x1 + 1000*x2 <= 50000',
        'x1 + x2 >= 30',
        'x2 >= 5',
        'x2 >= (1/3)*x1',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```