To solve the optimization problem described, we need to formulate a linear programming model that captures the constraints and objectives outlined. The goal is to maximize profit given the limitations on worker time and cardboard availability.

Let's denote:
- \(x\) as the number of glass jars packaged,
- \(y\) as the number of plates packaged.

The objective function to maximize profit is: 
\[2x + 2.5y\]

Given constraints are:
1. Worker time constraint: \(15x + 12y \leq 620\)
2. Cardboard availability constraint: \(3x + 4y \leq 120\)
3. Non-negativity constraints: \(x \geq 0, y \geq 0\)

This problem can be solved using Gurobi, a powerful linear and mixed-integer programming solver.

Here is the reasoning behind formulating this problem:
- The profit from each glass jar is $2, and from each plate is $2.50. Therefore, we aim to maximize \(2x + 2.5y\).
- Each glass jar requires 15 minutes of worker time, and each plate requires 12 minutes. With 620 minutes available, the total time used by both products must not exceed this limit.
- Similarly, for cardboard, with 3 units needed per glass jar and 4 units per plate, and only 120 units available, we have the constraint \(3x + 4y \leq 120\).

Now, let's implement this in Gurobi using Python:

```python
from gurobipy import *

# Create a new model
m = Model("Packaging_Optimization")

# Define variables
x = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="glass_jars")
y = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="plates")

# Set objective function
m.setObjective(2*x + 2.5*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(15*x + 12*y <= 620, "worker_time")
m.addConstr(3*x + 4*y <= 120, "cardboard")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x.varName} = {x.x}, {y.varName} = {y.x}")
    print(f"Maximum profit: ${2*x.x + 2.5*y.x:.2f}")
else:
    print("No optimal solution found")
```