To solve Sid's problem, we need to convert the natural language description into a symbolic representation of the optimization problem. Let's denote:

- $x_1$ as the number of hamburgers Sid buys
- $x_2$ as the number of plates of pasta Sid buys

The objective function is to minimize the total cost, which can be represented algebraically as:
\[3x_1 + 4x_2\]

Given the requirements and the nutritional content of each item, we have the following constraints:

1. Meat requirement: $x_1 \geq 2$ (since one hamburger has 1 serving of meat)
2. Dairy requirement: $0.5x_1 + x_2 \geq 1$
3. Vegetables requirement: $x_1 + x_2 \geq 4$
4. Grains requirement: $x_1 + 2x_2 \geq 3$

Since Sid cannot buy a negative number of items, we also have:
- $x_1 \geq 0$
- $x_2 \geq 0$

Thus, the symbolic representation of the problem is:

```json
{
  'sym_variables': [('x1', 'number of hamburgers'), ('x2', 'number of 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'
  ]
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a model
m = Model("Sid's Problem")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hamburgers")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="pasta")

# Set objective function
m.setObjective(3*x1 + 4*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(x1 >= 2, "meat_requirement")
m.addConstr(0.5*x1 + x2 >= 1, "dairy_requirement")
m.addConstr(x1 + x2 >= 4, "vegetables_requirement")
m.addConstr(x1 + 2*x2 >= 3, "grains_requirement")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Hamburgers: {x1.x}")
    print(f"Pasta: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```