## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The factory needs to decide how many glass jars and plates to package in order to maximize profit, given the constraints on worker time and cardboard availability.

Let's define the decision variables:

* `jars`: the number of glass jars to package
* `plates`: the number of plates to package

The objective function is to maximize profit:

* Profit per glass jar: $2
* Profit per plate: $2.50

The constraints are:

* Worker time: 15 minutes per glass jar, 12 minutes per plate, and 620 minutes available
* Cardboard: 3 units per glass jar, 4 units per plate, and 120 units available

## Formulation

The problem can be formulated as:

Maximize: 2 `jars` + 2.5 `plates`

Subject to:

* 15 `jars` + 12 `plates` <= 620 (worker time constraint)
* 3 `jars` + 4 `plates` <= 120 (cardboard constraint)
* `jars` >= 0, `plates` >= 0 (non-negativity constraints)

## Gurobi Code

```python
import gurobi

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

# Define the decision variables
jars = model.addVar(lb=0, name="jars", vtype=gurobi.GRB.CONTINUOUS)
plates = model.addVar(lb=0, name="plates", vtype=gurobi.GRB.CONTINUOUS)

# Define the objective function
model.setObjective(2 * jars + 2.5 * plates, gurobi.GRB.MAXIMIZE)

# Define the constraints
worker_time_constraint = model.addConstr(15 * jars + 12 * plates <= 620, name="worker_time_constraint")
cardboard_constraint = model.addConstr(3 * jars + 4 * plates <= 120, name="cardboard_constraint")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of jars: {jars.varValue}")
    print(f"Number of plates: {plates.varValue}")
    print(f"Max profit: {model.objVal}")
else:
    print("No optimal solution found.")
```

This code defines the decision variables, objective function, and constraints, and then solves the model using Gurobi's optimization algorithm. If an optimal solution is found, it prints the number of jars and plates to package and the maximum profit. Otherwise, it indicates that no optimal solution was found.