## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

- Let $x_1$ represent the number of 'ham sandwiches' and $x_2$ represent the number of 'corn cobs'.

## Step 2: Translate the problem into a symbolic optimization problem
The objective function to maximize is $5x_1 + 6x_2$.

## Step 3: List all constraints in symbolic form
Constraints:
1. $10.81x_1 + 1.92x_2 \geq 28$ (calcium constraint)
2. $11.85x_1 + 6.3x_2 \geq 22$ (umami index constraint)
3. $2.72x_1 + 8.06x_2 \geq 22$ (iron constraint)
4. $9x_1 - 2x_2 \geq 0$ (ham sandwiches and corn cobs relationship)
5. $10.81x_1 + 1.92x_2 \leq 38$ (calcium upper bound)
6. $11.85x_1 + 6.3x_2 \leq 61$ (umami index upper bound)
7. $2.72x_1 + 8.06x_2 \leq 44$ (iron upper bound)
8. $x_1 \in \mathbb{Z}$ (integer constraint for ham sandwiches)
9. $x_2 \in \mathbb{Z}$ (integer constraint for corn cobs)

## 4: Create the Gurobi model and variables
We will now create a Gurobi model and define the variables and constraints.

## 5: Implement the Gurobi code
```python
import gurobi

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

# Define the variables
x1 = model.addVar(name="ham_sandwiches", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(name="corn_cobs", vtype=gurobi.GRB.INTEGER)

# Define the objective function
model.setObjective(5 * x1 + 6 * x2, gurobi.GRB.MAXIMIZE)

# Add constraints
model.addConstr(10.81 * x1 + 1.92 * x2 >= 28, name="calcium_constraint")
model.addConstr(11.85 * x1 + 6.3 * x2 >= 22, name="umami_index_constraint")
model.addConstr(2.72 * x1 + 8.06 * x2 >= 22, name="iron_constraint")
model.addConstr(9 * x1 - 2 * x2 >= 0, name="ham_sandwiches_corn_cobs_relationship")
model.addConstr(10.81 * x1 + 1.92 * x2 <= 38, name="calcium_upper_bound")
model.addConstr(11.85 * x1 + 6.3 * x2 <= 61, name="umami_index_upper_bound")
model.addConstr(2.72 * x1 + 8.06 * x2 <= 44, name="iron_upper_bound")

# Optimize the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of ham sandwiches: {x1.varValue}")
    print(f"Number of corn cobs: {x2.varValue}")
    print(f"Objective function value: {model.objVal}")
else:
    print("No optimal solution found.")
```

## 6: Symbolic Representation
```json
{
    'sym_variables': [('x1', 'ham sandwiches'), ('x2', 'corn cobs')],
    'objective_function': '5*x1 + 6*x2',
    'constraints': [
        '10.81*x1 + 1.92*x2 >= 28',
        '11.85*x1 + 6.3*x2 >= 22',
        '2.72*x1 + 8.06*x2 >= 22',
        '9*x1 - 2*x2 >= 0',
        '10.81*x1 + 1.92*x2 <= 38',
        '11.85*x1 + 6.3*x2 <= 61',
        '2.72*x1 + 8.06*x2 <= 44',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ'
    ]
}
```