## Step 1: Define the symbolic representation of the problem
The problem involves two variables: 'bananas' and 'milkshakes', which we can represent symbolically as $x_1$ and $x_2$ respectively. The objective function to maximize is $9.92x_1x_2 + 7.8x_2^2 + 3.23x_1 + 6.31x_2$. The constraints are:
- $x_1 \geq 0$ and $x_2 \geq 0$ (implicit, as quantities cannot be negative)
- $1x_1 + 6x_2 \leq 47$ (total fat constraint)
- $x_1^2 + x_2^2 \geq 15$ (minimum fat from squares constraint)
- $7x_1^2 - 8x_2^2 \geq 0$ (relationship between bananas and milkshakes squared)
- $x_1 + x_2 \leq 36$ is mentioned twice but seems to be a repetition of a fat constraint, which we'll interpret as part of the fat constraint.

## 2: Correct and Clarify Constraints
Upon review, it seems there might have been a confusion in interpreting the constraints. The correct interpretation should be:
- The fat constraint from the attributes: $1x_1 + 6x_2 \leq 47$
- The minimum fat from squares: $x_1^2 + x_2^2 \geq 15$
- The relationship constraint: $7x_1^2 - 8x_2^2 \geq 0$
- And an upper bound on the sum of bananas and milkshakes: $x_1 + x_2 \leq 36$

## 3: Formulate the Symbolic Representation
The symbolic representation is:
- Variables: $(x_1, bananas), (x_2, milkshakes)$
- Objective Function: $9.92x_1x_2 + 7.8x_2^2 + 3.23x_1 + 6.31x_2$
- Constraints:
  1. $x_1 + 6x_2 \leq 47$
  2. $x_1^2 + x_2^2 \geq 15$
  3. $7x_1^2 - 8x_2^2 \geq 0$
  4. $x_1 + x_2 \leq 36$

## 4: Implement in Gurobi
To implement this in Gurobi, we'll use Python.

```python
import gurobi

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

# Define variables
bananas = model.addVar(lb=0, name="bananas", vtype=gurobi.GRB.CONTINUOUS)
milkshakes = model.addVar(lb=0, name="milkshakes", vtype=gurobi.GRB.CONTINUOUS)

# Objective function
model.setObjective(9.92 * bananas * milkshakes + 7.8 * milkshakes**2 + 3.23 * bananas + 6.31 * milkshakes, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(bananas + 6 * milkshakes <= 47, name="fat_constraint")
model.addConstr(bananas**2 + milkshakes**2 >= 15, name="min_fat_squares_constraint")
model.addConstr(7 * bananas**2 - 8 * milkshakes**2 >= 0, name="relationship_constraint")
model.addConstr(bananas + milkshakes <= 36, name="sum_constraint")

# Optimize
model.optimize()

# Print solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Bananas: {bananas.varValue}")
    print(f"Milkshakes: {milkshakes.varValue}")
    print(f"Objective: {model.objVal}")
else:
    print("No optimal solution found.")
```

## 5: Symbolic Representation Output
The symbolic representation in JSON format as requested:

```json
{
    'sym_variables': [('x1', 'bananas'), ('x2', 'milkshakes')],
    'objective_function': '9.92*x1*x2 + 7.8*x2^2 + 3.23*x1 + 6.31*x2',
    'constraints': [
        'x1 + 6*x2 <= 47',
        'x1^2 + x2^2 >= 15',
        '7*x1^2 - 8*x2^2 >= 0',
        'x1 + x2 <= 36'
    ]
}
```