Here's how we can formulate this problem and the corresponding Gurobi code:

**Decision Variables:**

* `x`: Number of hamburgers Sid eats.
* `y`: Number of plates of pasta Sid eats.

**Objective Function:**

Minimize the total cost:  `3x + 4y`

**Constraints:**

* **Meat:** `1x + 0y >= 2` (At least 2 servings of meat)
* **Dairy:** `0.5x + 1y >= 1` (At least 1 serving of dairy)
* **Vegetables:** `1x + 1y >= 4` (At least 4 servings of vegetables)
* **Grains:** `1x + 2y >= 3` (At least 3 servings of grains)
* **Non-negativity:** `x >= 0`, `y >= 0` (Can't eat negative amounts of food)


```python
import gurobipy as gp

# Create a new model
model = gp.Model("Sid's Diet")

# Create decision variables
x = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="hamburgers")
y = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="pasta")

# Set objective function
model.setObjective(3*x + 4*y, gp.GRB.MINIMIZE)

# Add constraints
model.addConstr(x >= 2, "meat")
model.addConstr(0.5*x + y >= 1, "dairy")
model.addConstr(x + y >= 4, "vegetables")
model.addConstr(x + 2*y >= 3, "grains")

# Optimize model
model.optimize()

# Print results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal cost: ${model.objVal}")
    print(f"Number of hamburgers: {x.x}")
    print(f"Number of pasta plates: {y.x}")
else:
    print("No solution found.")

```
