## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit from selling consoles and discs under given constraints.

### Variables
- \(x\): Number of consoles sold
- \(y\): Number of discs sold

### Objective Function
The profit from selling one console is $200, and from selling one disc is $30. Therefore, the total profit \(P\) can be represented as:
\[ P = 200x + 30y \]

### Constraints
1. **Budget Constraint**: The total cost for consoles and discs must not exceed $30,000. Given that a console costs $300 and a disc costs $30:
\[ 300x + 30y \leq 30000 \]

2. **Console Sales Constraint**: The number of consoles sold must be at least 20 but at most 50:
\[ 20 \leq x \leq 50 \]

3. **Disc Sales Constraint**: The number of discs sold is at most five times the number of consoles sold:
\[ y \leq 5x \]

4. **Non-Negativity Constraint**: The number of consoles and discs sold cannot be negative:
\[ x \geq 0, y \geq 0 \]
However, given the console sales constraint, \(x \geq 20\), which inherently satisfies \(x \geq 0\).

## Gurobi Code Formulation

```python
import gurobi

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

    # Define variables
    x = model.addVar(lb=20, ub=50, name="consoles")  # Number of consoles
    y = model.addVar(lb=0, name="discs")  # Number of discs

    # Objective function: Maximize profit
    model.setObjective(200 * x + 30 * y, gurobi.GRB.MAXIMIZE)

    # Budget constraint
    model.addConstr(300 * x + 30 * y <= 30000, name="budget_constraint")

    # Disc sales constraint
    model.addConstr(y <= 5 * x, name="disc_sales_constraint")

    # Solve the model
    model.optimize()

    # Check if the model is optimized
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Consoles = {x.varValue}, Discs = {y.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("The model is infeasible")

maximize_profit()
```

This code defines the optimization problem using Gurobi's Python interface, sets up the objective function and constraints according to the problem description, and solves the model to find the optimal number of consoles and discs to maximize profit.