To solve Gary's diet problem using linear programming, we need to define the decision variables, objective function, and constraints based on the given information.

Let's denote:
- \(x\) as the number of servings of noodles,
- \(y\) as the number of servings of cakes.

The objective is to minimize the total cost, which can be represented by the equation \(3x + 5y\), since noodles cost $3 per serving and cakes cost $5 per serving.

The constraints based on the nutritional requirements are:
1. Protein constraint: \(5x + 4y \leq 100\) (to receive at most 100 units of proteins),
2. Mineral constraint: \(12x + 8y \leq 30\) (to receive at most 30 units of minerals).

Additionally, since Gary cannot eat a negative number of servings, we have non-negativity constraints:
- \(x \geq 0\)
- \(y \geq 0\)

Now, let's translate this into Gurobi code in Python:

```python
from gurobipy import *

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

# Define the decision variables
x = m.addVar(vtype=GRB.CONTINUOUS, name="noodles")
y = m.addVar(vtype=GRB.CONTINUOUS, name="cakes")

# Set the objective function to minimize cost
m.setObjective(3*x + 5*y, GRB.MINIMIZE)

# Add constraints
m.addConstr(5*x + 4*y <= 100, "protein_constraint")
m.addConstr(12*x + 8*y <= 30, "mineral_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Servings of noodles: {x.x}")
    print(f"Servings of cakes: {y.x}")
    print(f"Total cost: ${3*x.x + 5*y.x:.2f}")
else:
    print("No optimal solution found")
```