To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the information provided.

Let's define:
- $x_1$ as the number of packages of LED lightbulbs to be made.
- $x_2$ as the number of packages of Halogen lightbulbs to be made.

The objective is to maximize profit. Given that the profit per package of LED lightbulbs is $30 and the profit per package of Halogen lightbulbs is $50, the objective function can be written as:
\[ \text{Maximize: } 30x_1 + 50x_2 \]

The constraints are based on the time availability of the machines. Each LED package requires 6 minutes on the plastics machine and 12 minutes on the wiring machine, while each Halogen package requires 9 minutes on the plastics machine and 10 minutes on the wiring machine. Both machines are available for at most 500 minutes per day. Thus, we have two constraints:
1. For the plastics machine: $6x_1 + 9x_2 \leq 500$
2. For the wiring machine: $12x_1 + 10x_2 \leq 500$

Additionally, since we cannot produce a negative number of packages, we have non-negativity constraints:
- $x_1 \geq 0$
- $x_2 \geq 0$

Now, let's represent this problem symbolically and then translate it into Gurobi code:

```json
{
    'sym_variables': [('x1', 'number of packages of LED lightbulbs'), ('x2', 'number of packages of Halogen lightbulbs')],
    'objective_function': '30*x1 + 50*x2',
    'constraints': ['6*x1 + 9*x2 <= 500', '12*x1 + 10*x2 <= 500', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we'll write the following code:

```python
from gurobipy import *

# Create a model
m = Model("Lightbulb_Production")

# Define variables
x1 = m.addVar(name="LED", vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name="Halogen", vtype=GRB.CONTINUOUS, lb=0)

# Set the objective function
m.setObjective(30*x1 + 50*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(6*x1 + 9*x2 <= 500, name="Plastics_Machine")
m.addConstr(12*x1 + 10*x2 <= 500, name="Wiring_Machine")

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")
print("Objective:", m.objVal)
```