## Step 1: Define the symbolic representation of the variables
The variables are 'bowls of instant ramen' and 'hot dogs', which we can denote as $x_1$ and $x_2$ respectively.

## Step 2: Translate the objective function into symbolic notation
The objective function to minimize is $6.45x_1x_2 + 2.48x_2$.

## 3: Define the constraints in symbolic notation
The constraints are:
1. $5x_1 + 6x_2 \geq 61$
2. $-6x_1 + 7x_2 \geq 0$
3. $5^2x_1^2 + 6^2x_2^2 \leq 122$ or $(5x_1)^2 + (6x_2)^2 \leq 122$ which simplifies to $25x_1^2 + 36x_2^2 \leq 122$
4. $x_2$ must be an integer.

## 4: Express the problem in the required JSON format
```json
{
'sym_variables': [('x1', 'bowls of instant ramen'), ('x2', 'hot dogs')], 
'objective_function': '6.45*x1*x2 + 2.48*x2', 
'constraints': [
    '5*x1 + 6*x2 >= 61', 
    '-6*x1 + 7*x2 >= 0', 
    '25*x1^2 + 36*x2^2 <= 122',
    'x2 is an integer'
]
}
```

## 5: Implement the optimization problem using Gurobi
To solve this problem, we will use Gurobi, an optimization solver. We need to define the variables, the objective function, and the constraints.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="bowls_of_instant_ramen", lb=0)  # No lower bound specified, assuming 0
x2 = m.addVar(name="hot_dogs", lb=0, vtype=gp.GRB.INTEGER)  # x2 must be an integer

# Define the objective function
m.setObjective(6.45 * x1 * x2 + 2.48 * x2, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(5 * x1 + 6 * x2 >= 61)
m.addConstr(-6 * x1 + 7 * x2 >= 0)
m.addConstr(25 * x1**2 + 36 * x2**2 <= 122)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Bowls of instant ramen: {x1.varValue}")
    print(f"Hot dogs: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```

The final answer is: 
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(name="bowls_of_instant_ramen", lb=0)  
x2 = m.addVar(name="hot_dogs", lb=0, vtype=gp.GRB.INTEGER)  

# Define the objective function
m.setObjective(6.45 * x1 * x2 + 2.48 * x2, gp.GRB.MINIMIZE)

# Define the constraints
m.addConstr(5 * x1 + 6 * x2 >= 61)
m.addConstr(-6 * x1 + 7 * x2 >= 0)
m.addConstr(25 * x1**2 + 36 * x2**2 <= 122)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Bowls of instant ramen: {x1.varValue}")
    print(f"Hot dogs: {x2.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```