To solve John's weight loss program problem, we first need to define the variables and constraints based on the given information. Let's denote:

- \(V\) as the number of plates of Vietnamese food.
- \(K\) as the number of plates of Korean food.

The objective is to minimize the total cost of the food while ensuring that the minimum requirements for protein and carbs are met.

Given:
- One plate of Vietnamese food contains 15 units of protein and 20 units of carbs, costing $12.5 per plate.
- One plate of Korean food contains 10 units of protein and 14 units of carbs, costing $16.5 per plate.
- Minimum requirements: 250 units of protein and 45 units of carbs.

We can formulate the problem as follows:

1. **Objective Function**: Minimize the total cost \(12.5V + 16.5K\).
2. **Constraints**:
   - Protein constraint: \(15V + 10K \geq 250\)
   - Carbs constraint: \(20V + 14K \geq 45\)
   - Non-negativity constraints: \(V \geq 0, K \geq 0\) (since we cannot have a negative number of plates).

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

```python
from gurobipy import *

# Create a model
m = Model("Weight_Loss_Program")

# Define variables
V = m.addVar(vtype=GRB.CONTINUOUS, name="Vietnamese_Food")
K = m.addVar(vtype=GRB.CONTINUOUS, name="Korean_Food")

# Set objective function
m.setObjective(12.5*V + 16.5*K, GRB.MINIMIZE)

# Add constraints
m.addConstr(15*V + 10*K >= 250, "Protein_Constraint")
m.addConstr(20*V + 14*K >= 45, "Carbs_Constraint")

# Optimize model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Vietnamese Food Plates: {V.x}")
    print(f"Korean Food Plates: {K.x}")
    print(f"Total Cost: ${12.5*V.x + 16.5*K.x:.2f}")
else:
    print("No optimal solution found.")
```

This code defines the problem in terms of Gurobi variables and constraints, optimizes it, and then prints out the number of plates for each type of food and the total cost if an optimal solution is found.