## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define the variables as follows:

- $x_1$ = number of new grad engineers
- $x_2$ = number of senior engineers

The objective is to minimize the weekly wage bill. The wage bill can be represented as $1000x_1 + 3000x_2$.

The constraints given are:
1. The weekly wage bill must be kept below $100,000: $1000x_1 + 3000x_2 \leq 100000$
2. The projects require a minimum of 50 engineers: $x_1 + x_2 \geq 50$
3. At least 10 engineers must be senior engineers: $x_2 \geq 10$
4. The number of senior engineers should be at least a third the number of new grad engineers: $x_2 \geq \frac{1}{3}x_1$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'number of new grad engineers'), ('x2', 'number of senior engineers')],
    'objective_function': '1000*x1 + 3000*x2',
    'constraints': [
        '1000*x1 + 3000*x2 <= 100000',
        'x1 + x2 >= 50',
        'x2 >= 10',
        'x2 >= (1/3)*x1'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

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

    # Objective function: minimize the wage bill
    model.setObjective(1000*x1 + 3000*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(1000*x1 + 3000*x2 <= 100000, name="wage_bill_constraint")
    model.addConstr(x1 + x2 >= 50, name="total_engineers_constraint")
    model.addConstr(x2 >= 10, name="senior_engineers_min_constraint")
    model.addConstr(x2 >= (1/3)*x1, name="senior_to_new_grad_ratio_constraint")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Number of new grad engineers: {x1.varValue}")
        print(f"Number of senior engineers: {x2.varValue}")
        print(f"Minimum wage bill: ${model.objVal:.2f}")
    else:
        print("The model is infeasible.")

solve_optimization_problem()
```