## Problem Description and Formulation

The problem is an optimization problem with a given objective function and constraints. The objective function to be minimized is:

\[ 5.19P^2 + 6.13PM + 5.47PB + 9.83PH + 7.3M^2 + 4.72MB + 3.82B^2 + 8.95H^2 + 6.59P + 6.2H \]

where $P$ represents potatoes, $M$ represents milkshakes, $B$ represents protein bars, and $H$ represents hot dogs.

The constraints are:

1. Potatoes have a tastiness rating of 3.
2. Milkshakes have a tastiness rating of 8.
3. Protein bars have a tastiness rating of 1.
4. Hot dogs have a tastiness rating of 4.
5. $3^2P + 1^2B \geq 30$
6. $8M + 4H \geq 28$
7. $3P + 8M + 1B + 4H \geq 28$
8. $3P + 1B \leq 61$
9. $3P + 8M \leq 62$
10. $3P + 4H \leq 127$

## Gurobi Code Formulation

```python
import gurobi

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

    # Define the variables
    P = model.addVar(lb=0, name="potatoes", vtype=gurobi.GRB.CONTINUOUS)
    M = model.addVar(lb=0, name="milkshakes", vtype=gurobi.GRB.CONTINUOUS)
    B = model.addVar(lb=0, name="protein_bars", vtype=gurobi.GRB.CONTINUOUS)
    H = model.addVar(lb=0, name="hot_dogs", vtype=gurobi.GRB.CONTINUOUS)

    # Define the objective function
    model.setObjective(5.19*P**2 + 6.13*P*M + 5.47*P*B + 9.83*P*H + 
                       7.3*M**2 + 4.72*M*B + 3.82*B**2 + 8.95*H**2 + 
                       6.59*P + 6.2*H, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstraint(3**2 * P + 1**2 * B >= 30, name="tastiness_rating_1")
    model.addConstraint(8 * M + 4 * H >= 28, name="tastiness_rating_2")
    model.addConstraint(3 * P + 8 * M + 1 * B + 4 * H >= 28, name="tastiness_rating_3")
    model.addConstraint(3 * P + 1 * B <= 61, name="tastiness_rating_4")
    model.addConstraint(3 * P + 8 * M <= 62, name="tastiness_rating_5")
    model.addConstraint(3 * P + 4 * H <= 127, name="tastiness_rating_6")

    # Update model
    model.update()

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal Solution:")
        print(f"Potatoes: {P.varValue}")
        print(f"Milkshakes: {M.varValue}")
        print(f"Protein Bars: {B.varValue}")
        print(f"Hot Dogs: {H.varValue}")
        print(f"Objective Function Value: {model.objVal}")
    else:
        print("The model is infeasible.")

optimize_problem()
```