## Problem Description and Formulation

The farmer has 30 acres available for planting carrots and beets. The goal is to maximize profit given the following constraints:

1. The farmer must plant a minimum of 3 acres of carrots and 5 acres of beets.
2. The farmer prefers to plant more beets than carrots but can plant at most 2 times the quantity of beets as carrots.
3. The total acres used for carrots and beets cannot exceed 30 acres.
4. The profit per acre of carrots is $500, and the profit per acre of beets is $400.

Let's denote:
- \(C\) as the acres of carrots planted,
- \(B\) as the acres of beets planted.

The objective function to maximize profit (\(P\)) is:
\[ P = 500C + 400B \]

Subject to:
1. \( C \geq 3 \)
2. \( B \geq 5 \)
3. \( B \leq 2C \)
4. \( C + B \leq 30 \)
5. \( C, B \geq 0 \) (Non-negativity constraint, though the minimums ensure this)

## Gurobi Code

```python
import gurobi

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

    # Define variables
    C = m.addVar(lb=3, name="Carrots")  # Minimum 3 acres of carrots
    B = m.addVar(lb=5, name="Beets")    # Minimum 5 acres of beets

    # Objective function: Maximize profit
    m.setObjective(500*C + 400*B, gurobi.GRB.MAXIMIZE)

    # Constraints
    m.addConstr(C + B <= 30, name="Total_Acres")  # Total acres constraint
    m.addConstr(B <= 2*C, name="Beets_to_Carrots_Ratio")  # Beets to carrots ratio

    # Solve the model
    m.optimize()

    # Print solution
    if m.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal acres of carrots: {C.varValue}")
        print(f"Optimal acres of beets: {B.varValue}")
        print(f"Max Profit: {m.objVal}")
    else:
        print("The model is infeasible")

solve_farmer_problem()
```

This Gurobi code models the farmer's optimization problem, solving for the acres of carrots and beets that maximize profit under the given constraints.