## Problem Description and Formulation

The problem is a linear programming optimization problem. We need to minimize the cost of mixing animal feeds, specifically oats and sunflower seeds, while ensuring the mixture contains a minimum of 250 units of protein and 400 units of fat.

## Decision Variables

Let $x$ be the amount of oats in kilograms and $y$ be the amount of sunflower seeds in kilograms.

## Objective Function

The objective is to minimize the total cost of the mixture, which is given by $50x + 70y$.

## Constraints

1. **Protein Constraint**: The mixture must contain at least 250 units of protein. Since oats contain 5 units of protein per kilogram and sunflower seeds contain 10 units of protein per kilogram, we have $5x + 10y \geq 250$.
2. **Fat Constraint**: The mixture must contain at least 400 units of fat. Since oats contain 16 units of fat per kilogram and sunflower seeds contain 22 units of fat per kilogram, we have $16x + 22y \geq 400$.
3. **Non-Negativity Constraints**: The amounts of oats and sunflower seeds cannot be negative, so $x \geq 0$ and $y \geq 0$.

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    x = model.addVar(name="oats", lb=0, vtype=gurobi.GRB.CONTINUOUS)  # Amount of oats in kilograms
    y = model.addVar(name="sunflower_seeds", lb=0, vtype=gurobi.GRB.CONTINUOUS)  # Amount of sunflower seeds in kilograms

    # Define the objective function: Minimize the total cost
    model.setObjective(50 * x + 70 * y, gurobi.GRB.MINIMIZE)

    # Add the protein constraint: 5x + 10y >= 250
    model.addConstr(5 * x + 10 * y >= 250, name="protein_constraint")

    # Add the fat constraint: 16x + 22y >= 400
    model.addConstr(16 * x + 22 * y >= 400, name="fat_constraint")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Amount of oats: {x.varValue} kg")
        print(f"Amount of sunflower seeds: {y.varValue} kg")
        print(f"Total cost: ${model.objVal:.2f}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible.")
    else:
        print("The model status is not optimal or infeasible.")

solve_animal_feed_problem()
```