## Problem Description and Formulation

The problem is a classic example of a linear programming problem. The goal is to maximize the profit of producing pasta sauce and barbecue sauce given the constraints of machine availability.

Let's define the variables:
- \(x\): the number of jars of pasta sauce to be produced
- \(y\): the number of jars of barbecue sauce to be produced

The objective function to maximize profit is:
\[ \text{Maximize:} \quad 3x + 5y \]

The constraints based on machine availability are:
- Filling machine: \( x + 3y \leq 12500 \)
- Jarring machine: \( 3x + 4y \leq 20000 \)
- Non-negativity: \( x \geq 0, y \geq 0 \)

## Gurobi Code

To solve this problem using Gurobi in Python, we will use the Gurobi Python API. First, ensure you have Gurobi installed in your environment. You can install it via pip:
```bash
pip install gurobi
```

Here's the Gurobi code to solve the problem:

```python
import gurobi as gp

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

# Define variables
x = model.addVar(name="pasta_sauce", lb=0, vtype=gp.GRB.CONTINUOUS)
y = model.addVar(name="barbecue_sauce", lb=0, vtype=gp.GRB.CONTINUOUS)

# Objective function: Maximize 3x + 5y
model.setObjective(3*x + 5*y, sense=gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(x + 3*y <= 12500, name="filling_machine_constraint")
model.addConstr(3*x + 4*y <= 20000, name="jarring_machine_constraint")

# Optimize the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal production levels: Pasta Sauce = {x.varValue}, Barbecue Sauce = {y.varValue}")
    print(f"Maximum profit: ${3*x.varValue + 5*y.varValue:.2f}")
else:
    print("The model is infeasible or unbounded.")
```