## Problem Description and Formulation

The problem is to maximize the number of oranges an orange farm can send to the city given a budget of $500. The farm can use two modes of transportation: train and car. Each train trip costs $50 and can carry 500 oranges, while each car trip costs $30 and can carry 200 oranges. Additionally, the number of train trips cannot exceed the number of car trips.

## Decision Variables

Let \(T\) be the number of train trips and \(C\) be the number of car trips.

## Objective Function

The objective is to maximize the total number of oranges sent, which is \(500T + 200C\).

## Constraints

1. **Budget Constraint**: The total cost must not exceed $500. The cost of train trips is $50\(T\) and the cost of car trips is $30\(C\), so we have \(50T + 30C \leq 500\).
2. **Train and Car Trips Constraint**: The number of train trips cannot exceed the number of car trips, so \(T \leq C\).
3. **Non-Negativity Constraint**: The number of trips cannot be negative, so \(T \geq 0\) and \(C \geq 0\).
4. **Integer Constraint**: Since the number of trips must be a whole number, \(T\) and \(C\) must be integers.

## Gurobi Code

```python
import gurobi

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

    # Define the decision variables
    T = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="Train Trips")
    C = model.addVar(lb=0, vtype=gurobi.GRB.INTEGER, name="Car Trips")

    # Objective function: Maximize the number of oranges sent
    model.setObjective(500*T + 200*C, gurobi.GRB.MAXIMIZE)

    # Budget constraint
    model.addConstr(50*T + 30*C <= 500, name="Budget Constraint")

    # Train trips cannot exceed car trips
    model.addConstr(T <= C, name="Train Car Trips Constraint")

    # Solve the model
    model.optimize()

    if model.status == gurobi.GRB.Status.OPTIMAL:
        print(f"Optimal Solution: Train Trips = {T.varValue}, Car Trips = {C.varValue}")
        print(f"Maximum Oranges Sent: {500*T.varValue + 200*C.varValue}")
    else:
        print("The model is infeasible")

solve_orange_farm_problem()
```