To solve Luke's problem, we first need to translate the natural language description into a symbolic representation of the optimization problem. Let's denote the number of hectares of carrots as $x_1$ and the number of hectares of pumpkins as $x_2$. The objective is to maximize profit, with carrots generating $300 per hectare and pumpkins generating $500 per hectare.

The symbolic representation of the problem can be encapsulated in a JSON object as follows:
```json
{
    'sym_variables': [('x1', 'hectares of carrots'), ('x2', 'hectares of pumpkins')],
    'objective_function': '300*x1 + 500*x2',
    'constraints': [
        'x1 + x2 <= 200', 
        'x1 >= 25', 
        'x2 >= 20', 
        'x1 <= 2*x2'
    ]
}
```
This representation captures the essence of the problem: maximizing profit (`300*x1 + 500*x2`) under the given constraints:
- The total area used for carrots and pumpkins cannot exceed 200 hectares (`x1 + x2 <= 200`).
- At least 25 hectares must be dedicated to carrots (`x1 >= 25`).
- At least 20 hectares must be dedicated to pumpkins (`x2 >= 20`).
- The amount of land used for carrots cannot exceed twice the amount used for pumpkins (`x1 <= 2*x2`).

Given this symbolic representation, we can now formulate the Gurobi code in Python to solve the optimization problem:
```python
from gurobipy import *

# Create a new model
m = Model("Luke_Farming_Problem")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="hectares_of_carrots")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="hectares_of_pumpkins")

# Set the objective function
m.setObjective(300*x1 + 500*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(x1 + x2 <= 200, "total_area")
m.addConstr(x1 >= 25, "min_carrots")
m.addConstr(x2 >= 20, "min_pumpkins")
m.addConstr(x1 <= 2*x2, "carrot_to_pumpkin_ratio")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: ${m.objVal}")
else:
    print("No optimal solution found")
```