## Problem Description and Symbolic Representation

The problem requires finding the minimum cost for Sid to meet his daily requirements of meat, dairy, vegetables, and grains by purchasing hamburgers and plates of pasta.

### Symbolic Representation:

Let's denote:
- $x_1$ as the number of hamburgers
- $x_2$ as the number of plates of pasta

The objective function to minimize is the total cost: $3x_1 + 4x_2$

The constraints based on the daily requirements are:
- Meat: $x_1 \geq 2$
- Dairy: $0.5x_1 + x_2 \geq 1$
- Vegetables: $x_1 + x_2 \geq 4$
- Grains: $x_1 + 2x_2 \geq 3$
- Non-negativity: $x_1 \geq 0, x_2 \geq 0$

Since $x_1$ and $x_2$ represent the number of items, they should be integers.

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'hamburgers'), ('x2', 'plates of pasta')],
    'objective_function': '3*x1 + 4*x2',
    'constraints': [
        'x1 >= 2',
        '0.5*x1 + x2 >= 1',
        'x1 + x2 >= 4',
        'x1 + 2*x2 >= 3',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="hamburgers", vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="plates_of_pasta", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(3*x1 + 4*x2, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(x1 >= 2, name="meat_requirement")
    model.addConstr(0.5*x1 + x2 >= 1, name="dairy_requirement")
    model.addConstr(x1 + x2 >= 4, name="vegetables_requirement")
    model.addConstr(x1 + 2*x2 >= 3, name="grains_requirement")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of hamburgers: {x1.varValue}")
        print(f"Number of plates of pasta: {x2.varValue}")
        print(f"Minimum cost: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```