To solve this optimization problem using Gurobi, we first need to understand and translate the given natural language description into a mathematical formulation. The objective is to minimize the function \(6.45 \times (\text{bowls of instant ramen}) \times (\text{hot dogs}) + 2.48 \times (\text{hot dogs})\), subject to several constraints related to tastiness ratings and other conditions.

Let's denote:
- \(x_0\) as the number of bowls of instant ramen,
- \(x_1\) as the number of hot dogs.

The objective function to minimize is: \(6.45x_0x_1 + 2.48x_1\).

Constraints are:
1. Tastiness rating constraint: \(5x_0 + 6x_1 \geq 61\).
2. Since the tastiness must be at least 61, this constraint already covers the requirement for being equal to or greater than 61.
3. Linear constraint: \(-6x_0 + 7x_1 \geq 0\).
4. Quadratic constraint on tastiness ratings squared: \(25x_0^2 + 36x_1^2 \leq 122\).

Given that \(x_0\) (bowls of instant ramen) can be a non-integer but \(x_1\) (hot dogs) must be an integer, we model \(x_0\) as a continuous variable and \(x_1\) as an integer variable.

Here is the Gurobi code to solve this optimization problem:

```python
from gurobipy import *

# Create a new model
m = Model("Optimization_Problem")

# Define variables
x0 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.CONTINUOUS, name="bowls_of_instant_ramen")
x1 = m.addVar(lb=0, ub=GRB.INFINITY, vtype=GRB.INTEGER, name="hot_dogs")

# Objective function
m.setObjective(6.45*x0*x1 + 2.48*x1, GRB.MINIMIZE)

# Constraints
m.addConstr(5*x0 + 6*x1 >= 61, "tastiness_rating_constraint")
m.addConstr(-6*x0 + 7*x1 >= 0, "linear_constraint")
m.addConstr(25*x0**2 + 36*x1**2 <= 122, "quadratic_tastiness_constraint")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of bowls of instant ramen: {x0.x}")
    print(f"Number of hot dogs: {int(x1.x)}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```