To solve this optimization problem, we first need to define the variables and constraints based on the given information. Let's denote:

- \(x\) as the number of jars of pasta sauce produced.
- \(y\) as the number of jars of barbecue sauce produced.

The objective is to maximize profit, which can be calculated as $3x + 5y$ since each jar of pasta sauce gives a profit of $3 and each jar of barbecue sauce gives a profit of $5.

There are two main constraints based on the availability of machines:

1. **Filling Machine Constraint**: Each jar of pasta sauce takes 1 minute, and each jar of barbecue sauce takes 3 minutes on the filling machine. The total time available on the filling machine is 12,500 minutes. Therefore, the constraint can be represented as:
\[x + 3y \leq 12500\]

2. **Jarring Machine Constraint**: Each jar of pasta sauce takes 3 minutes, and each jar of barbecue sauce takes 4 minutes on the jarring machine. The total time available on the jarring machine is 20,000 minutes. Therefore, the constraint can be represented as:
\[3x + 4y \leq 20000\]

Additionally, \(x\) and \(y\) must be non-negative since they represent quantities of jars.

Now, let's write the Gurobi code to solve this linear programming problem:

```python
from gurobipy import *

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

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

# Set the objective function: Maximize profit
m.setObjective(3*x + 5*y, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x + 3*y <= 12500, "filling_machine_constraint")
m.addConstr(3*x + 4*y <= 20000, "jarring_machine_constraint")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of jars of pasta sauce: {x.x}")
    print(f"Number of jars of barbecue sauce: {y.x}")
    print(f"Maximum profit: ${3*x.x + 5*y.x:.2f}")
else:
    print("No optimal solution found")
```