## Problem Description and Symbolic Representation

The problem involves a butcher shop that needs to determine the optimal number of burgers and sausages to produce given certain constraints. The goal is to maximize profit.

### Symbolic Variables:
- Let \(x_1\) represent the number of burgers.
- Let \(x_2\) represent the number of sausages.

### Objective Function:
The profit from each burger is $5, and from each sausage is $3. Therefore, the objective function to maximize profit (\(P\)) is:
\[ P = 5x_1 + 3x_2 \]

### Constraints:
1. **Total Meat Constraint**: The shop has 1000 grams of ground meat. Each burger requires 20 grams, and each sausage requires 10 grams.
\[ 20x_1 + 10x_2 \leq 1000 \]

2. **Sausages to Burgers Ratio Constraint**: At least three times the number of sausages are needed than burgers.
\[ x_2 \geq 3x_1 \]

3. **Minimum Burgers Constraint**: There needs to be at least 10 burgers made.
\[ x_1 \geq 10 \]

4. **Non-Negativity Constraints**: The number of burgers and sausages cannot be negative.
\[ x_1 \geq 0, x_2 \geq 0 \]

However, since \(x_1 \geq 10\) is specified, the \(x_1 \geq 0\) and \(x_2 \geq 0\) constraints are implicitly satisfied.

## Symbolic Representation in JSON Format

```json
{
  'sym_variables': [('x1', 'burgers'), ('x2', 'sausages')],
  'objective_function': '5*x1 + 3*x2',
  'constraints': [
    '20*x1 + 10*x2 <= 1000',
    'x2 >= 3*x1',
    'x1 >= 10'
  ]
}
```

## Gurobi Code in Python

```python
import gurobi

def butcher_shop_optimization():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name="burgers", lb=0, vtype=gurobi.GRB.INTEGER)  # Number of burgers
    x2 = model.addVar(name="sausages", lb=0, vtype=gurobi.GRB.INTEGER)  # Number of sausages

    # Objective function: Maximize profit
    model.setObjective(5 * x1 + 3 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(20 * x1 + 10 * x2 <= 1000, name="meat_constraint")  # Total meat constraint
    model.addConstr(x2 >= 3 * x1, name="ratio_constraint")  # Sausages to burgers ratio constraint
    model.addConstr(x1 >= 10, name="min_burgers_constraint")  # Minimum burgers constraint

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal number of burgers: {x1.varValue}")
        print(f"Optimal number of sausages: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible.")

butcher_shop_optimization()
```