To solve Sid's problem, we need to formulate a linear programming model. The goal is to minimize the total cost of meeting the daily requirements of meat, dairy, vegetables, and grains.

Let's define the variables:
- $x_1$ = number of hamburgers
- $x_2$ = number of plates of pasta

The objective function represents the total cost that Sid wants to minimize. The cost of a hamburger is $3, and the cost of a plate of pasta is $4. Therefore, the objective function can be written as:
\[ \text{Minimize} \quad 3x_1 + 4x_2 \]

The constraints are based on the nutritional requirements and the composition of each food item:
- Meat requirement: $1x_1 + 0x_2 \geq 2$ (since a hamburger has 1 serving of meat and pasta has 0 servings)
- Dairy requirement: $0.5x_1 + 1x_2 \geq 1$ (since a hamburger has 0.5 servings of dairy and pasta has 1 serving)
- Vegetables requirement: $1x_1 + 1x_2 \geq 4$ (since both hamburger and pasta have 1 serving of vegetables)
- Grains requirement: $1x_1 + 2x_2 \geq 3$ (since a hamburger has 1 serving of grains and pasta has 2 servings)

Additionally, we cannot buy negative quantities of food, so:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

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

```python
from gurobipy import *

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

# Define the variables
x1 = m.addVar(name='hamburgers', vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name='pasta', vtype=GRB.CONTINUOUS, lb=0)

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

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

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print(f"Objective Function Value: {m.objVal}")
```
```python
from gurobipy import *

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

# Define the variables
x1 = m.addVar(name='hamburgers', vtype=GRB.CONTINUOUS, lb=0)
x2 = m.addVar(name='pasta', vtype=GRB.CONTINUOUS, lb=0)

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

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

# Optimize the model
m.optimize()

# Print the results
for v in m.getVars():
    print(f"{v.varName}: {v.x}")

print(f"Objective Function Value: {m.objVal}")
```