## Problem Description and Formulation

The problem is a linear programming optimization problem. Gary needs to minimize his cost of consuming noodles and cakes while adhering to the constraints on proteins and minerals intake.

### Decision Variables
- Let \(N\) be the number of servings of noodles Gary eats.
- Let \(C\) be the number of servings of cakes Gary eats.

### Objective Function
The objective is to minimize the total cost, which is given by \(3N + 5C\), as noodles cost $3 per serving and cakes cost $5 per serving.

### Constraints
1. **Proteins Constraint**: A serving of noodles contains 5 units of proteins, and a serving of cake contains 4 units of proteins. Gary wants to receive at most 100 units of proteins. This can be represented as \(5N + 4C \leq 100\).
2. **Minerals Constraint**: A serving of noodles contains 12 units of minerals, and a serving of cake contains 8 units of minerals. Gary wants to receive at most 30 units of minerals. This can be represented as \(12N + 8C \leq 30\).
3. **Non-Negativity Constraint**: The number of servings of noodles and cakes cannot be negative, i.e., \(N \geq 0\) and \(C \geq 0\).

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    N = model.addVar(name="noodles", lb=0, ub=gurobi.GRB.INFINITY)
    C = model.addVar(name="cakes", lb=0, ub=gurobi.GRB.INFINITY)

    # Define the objective function
    model.setObjective(3*N + 5*C, gurobi.GRB.MINIMIZE)

    # Add constraints
    model.addConstr(5*N + 4*C <= 100, name="proteins_constraint")
    model.addConstr(12*N + 8*C <= 30, name="minerals_constraint")

    # Optimize the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.Status.OPTIMAL:
        print("Optimal Solution:")
        print(f"Noodles: {N.varValue}")
        print(f"Cakes: {C.varValue}")
        print(f"Cost: {model.objVal}")
    else:
        print("The model is infeasible or unbounded.")

solve_diet_problem()
```

This code defines the optimization problem using Gurobi's Python interface, solves it, and prints out the optimal number of servings of noodles and cakes that Gary should eat to minimize his cost while adhering to the given constraints.