To solve the given optimization problem, we need to convert the natural language description into a symbolic representation that can be translated into Gurobi code in Python. The problem involves deciding how many batches of sausages and burger patties should be produced to maximize profit, subject to constraints on machine usage.

Let's denote:
- \(x_s\) as the number of batches of sausages produced.
- \(x_b\) as the number of batches of burger patties produced.

From the problem description, we have the following constraints and objective function:

1. **Objective Function**: Maximize profit.
   - Profit per batch of sausages = $200
   - Profit per batch of burger patties = $250
   - Total profit = \(200x_s + 250x_b\)

2. **Constraints**:
   - Meat-grinder constraint: \(2x_s + 4x_b \leq 3000\) (since each batch of sausages requires 2 hours and each batch of burger patties requires 4 hours, and the meat-grinder runs for at most 3000 hours per year).
   - Meat-packer constraint: \(3x_s + 1.5x_b \leq 3000\) (since each batch of sausages requires 3 hours and each batch of burger patties requires 1.5 hours, and the meat-packer runs for at most 3000 hours per year).
   - Non-negativity constraints: \(x_s \geq 0\), \(x_b \geq 0\) (since the number of batches cannot be negative).

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

```python
from gurobipy import *

# Create a model
m = Model("Sausage_and_Burger_Patty_Production")

# Define variables
x_s = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="sausages")
x_b = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="burger_patties")

# Set objective function
m.setObjective(200*x_s + 250*x_b, GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x_s + 4*x_b <= 3000, "meat_grinder_constraint")
m.addConstr(3*x_s + 1.5*x_b <= 3000, "meat_packer_constraint")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of batches of sausages: {x_s.x}")
    print(f"Number of batches of burger patties: {x_b.x}")
    print(f"Total profit: ${200*x_s.x + 250*x_b.x:.2f}")
else:
    print("No optimal solution found")
```