To solve the optimization problem described, we first need to translate the natural language description into a mathematical formulation that can be solved using linear programming techniques. The goal is to maximize profit given the constraints on the availability of ground meat and the demand for burgers and sausages.

Let's denote:
- \(B\) as the number of burgers made,
- \(S\) as the number of sausages made.

Given information:
- Total ground meat available = 1000 grams
- Ground meat required per burger = 20 grams
- Ground meat required per sausage = 10 grams
- Profit per burger = $5
- Profit per sausage = $3
- Minimum burgers to be made = 10
- Minimum sausages required is three times the number of burgers, so \(S \geq 3B\)

Objective: Maximize profit

The objective function can be written as:
\[ \text{Maximize} \quad 5B + 3S \]

Constraints:
1. Ground meat constraint: \(20B + 10S \leq 1000\)
2. Minimum burgers: \(B \geq 10\)
3. Sausage to burger ratio: \(S \geq 3B\)
4. Non-negativity constraints: \(B \geq 0, S \geq 0\)

Now, let's translate this formulation into Gurobi code in Python:

```python
from gurobipy import *

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

# Define variables
B = m.addVar(lb=0, vtype=GRB.INTEGER, name="burgers")
S = m.addVar(lb=0, vtype=GRB.INTEGER, name="sausages")

# Objective function: Maximize profit
m.setObjective(5*B + 3*S, GRB.MAXIMIZE)

# Constraints
m.addConstr(20*B + 10*S <= 1000, "ground_meat_constraint")
m.addConstr(B >= 10, "minimum_burgers")
m.addConstr(S >= 3*B, "sausage_to_burger_ratio")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {B.varName} = {int(B.x)}, {S.varName} = {int(S.x)}")
    print(f"Maximum Profit: ${5*int(B.x) + 3*int(S.x)}")
else:
    print("Model is infeasible")
```