## 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 glass jars packaged
- \(x_2\) as the number of plates packaged

The objective is to maximize profit. Given that the profit per glass jar is $2 and per plate is $2.50, the objective function can be written as:

\[ \text{Maximize:} \quad 2x_1 + 2.5x_2 \]

The constraints based on the given resources are:

1. Worker time: \(15x_1 + 12x_2 \leq 620\)
2. Cardboard: \(3x_1 + 4x_2 \leq 120\)
3. Non-negativity: \(x_1 \geq 0, x_2 \geq 0\)

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'glass jars'), ('x2', 'plates')],
    'objective_function': '2*x1 + 2.5*x2',
    'constraints': [
        '15*x1 + 12*x2 <= 620',
        '3*x1 + 4*x2 <= 120',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define variables
x1 = model.addVar(name="glass_jars", lb=0, vtype=gp.GRB.CONTINUOUS)
x2 = model.addVar(name="plates", lb=0, vtype=gp.GRB.CONTINUOUS)

# Objective function: Maximize 2*x1 + 2.5*x2
model.setObjective(2*x1 + 2.5*x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(15*x1 + 12*x2 <= 620, name="worker_time")
model.addConstr(3*x1 + 4*x2 <= 120, name="cardboard")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: x1 = {x1.varValue}, x2 = {x2.varValue}")
    print(f"Max Profit: {model.objVal}")
else:
    print("The model is infeasible")
```