## Symbolic Representation

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

- \(x_1\) as the number of meat lunches
- \(x_2\) as the number of veggie lunches

The objective is to maximize profit, where the profit per meat lunch is $8 and per veggie lunch is $6. So, the objective function can be represented as:

\[ \text{Maximize:} \quad 8x_1 + 6x_2 \]

The constraints based on the given information are:

1. Preparation time: \(5x_1 + 4x_2 \leq 500\)
2. Packaging time: \(3x_1 + 5x_2 \leq 400\)
3. Non-negativity: \(x_1 \geq 0, x_2 \geq 0\)

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'meat lunches'), ('x2', 'veggie lunches')],
    'objective_function': '8*x1 + 6*x2',
    'constraints': [
        '5*x1 + 4*x2 <= 500',
        '3*x1 + 5*x2 <= 400',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Lunch_Preparation")

# Define variables
x1 = model.addVar(name="meat_lunches", lb=0, vtype=gp.GRB.INTEGER)  # Number of meat lunches
x2 = model.addVar(name="veggie_lunches", lb=0, vtype=gp.GRB.INTEGER)  # Number of veggie lunches

# Objective function: Maximize profit
model.setObjective(8*x1 + 6*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(5*x1 + 4*x2 <= 500, name="prep_time")  # Preparation time constraint
model.addConstr(3*x1 + 5*x2 <= 400, name="packaging_time")  # Packaging time constraint

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: {model.x[0]} meat lunches, {model.x[1]} veggie lunches")
    print(f"Max Profit: ${model.objVal}")
else:
    print("No optimal solution found")
```