## Step 1: Define the symbolic representation of the variables
The variables are 'cheeseburgers' and 'hot dogs', which can be represented symbolically as $x_1$ and $x_2$ respectively.

## Step 2: Convert the natural language description into a symbolic representation
The objective function to minimize is $3x_1 + 6x_2$.

## Step 3: List all the constraints in symbolic notation
The constraints are:
- $5x_1 + 4x_2 \geq 11$ (at least 11 grams of fat from cheeseburgers and hot dogs)
- $2x_1 + 3x_2 \geq 8$ (at least 8 milligrams of calcium from cheeseburgers and hot dogs)
- $10x_1 - 3x_2 \geq 0$ (constraint on the number of cheeseburgers and hot dogs)
- $5x_1 + 4x_2 \leq 40$ (no more than 40 grams of fat from cheeseburgers and hot dogs)
- $2x_1 + 3x_2 \leq 13$ (up to 13 milligrams of calcium from cheeseburgers and hot dogs)
- $x_2 \geq 0$ and $x_2$ is an integer (whole number of hot dogs)
- $x_1 \geq 0$ (non-negativity constraint for cheeseburgers, and it can be fractional)

## 4: Define the symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'cheeseburgers'), ('x2', 'hot dogs')],
'objective_function': '3*x1 + 6*x2',
'constraints': [
    '5*x1 + 4*x2 >= 11',
    '2*x1 + 3*x2 >= 8',
    '10*x1 - 3*x2 >= 0',
    '5*x1 + 4*x2 <= 40',
    '2*x1 + 3*x2 <= 13',
    'x1 >= 0',
    'x2 >= 0 and x2 is an integer'
]
}
```

## 5: Implement the optimization problem using Gurobi
```python
import gurobi

# Create a new model
m = gurobi.Model()

# Define the variables
x1 = m.addVar(name="cheeseburgers", lb=0)  # cheeseburgers can be fractional
x2 = m.addVar(name="hot dogs", lb=0, integrality=gurobi.GRB.INTEGER)  # hot dogs must be an integer

# Objective function: minimize 3*cheeseburgers + 6*hot dogs
m.setObjective(3*x1 + 6*x2, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(5*x1 + 4*x2 >= 11, name="fat_requirement")
m.addConstr(2*x1 + 3*x2 >= 8, name="calcium_requirement")
m.addConstr(10*x1 - 3*x2 >= 0, name="cheeseburgers_vs_hotdogs")
m.addConstr(5*x1 + 4*x2 <= 40, name="fat_limit")
m.addConstr(2*x1 + 3*x2 <= 13, name="calcium_limit")

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Cheeseburgers: ", x1.varValue)
    print("Hot dogs: ", x2.varValue)
else:
    print("The model is infeasible")
```