## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit of a toy store that sells handmade wooden trains and planes, given certain constraints.

### Decision Variables

- Let \(T\) be the number of trains to be made.
- Let \(P\) be the number of planes to be made.

### Objective Function

The profit per train is $50, and the profit per plane is $60. The objective is to maximize the total profit:

\[ \text{Maximize:} \quad 50T + 60P \]

### Constraints

1. **Woodworker Time Constraint:** Each train takes 30 minutes of woodworker time, and each plane takes 40 minutes. The store has 4000 minutes of woodworker time available.

\[ 30T + 40P \leq 4000 \]

2. **Planes vs. Trains Constraint:** The store must make at least thrice the number of planes as trains.

\[ P \geq 3T \]

3. **Non-Negativity Constraint:** The number of trains and planes cannot be negative.

\[ T \geq 0, P \geq 0 \]

## Gurobi Code

To solve this problem using Gurobi in Python, we can use the following code:

```python
import gurobi

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

    # Define the decision variables
    T = model.addVar(lb=0, name="T")  # Number of trains
    P = model.addVar(lb=0, name="P")  # Number of planes

    # Objective function: Maximize profit
    model.setObjective(50 * T + 60 * P, gurobi.GRB.MAXIMIZE)

    # Woodworker time constraint
    model.addConstr(30 * T + 40 * P <= 4000, name="woodworker_time")

    # Planes vs. trains constraint
    model.addConstr(P >= 3 * T, name="planes_vs_trains")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution found.")
        print(f"Number of trains: {T.varValue}")
        print(f"Number of planes: {P.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible or unbounded.")

solve_toy_store_problem()
```

This code defines the optimization problem as described, sets up the model with Gurobi, and solves it. The solution will provide the number of trains and planes that maximize profit under the given constraints.