## Step 1: Define the symbolic representation of the problem
Let's denote the number of heated rooms as $x_1$ and the number of unheated rooms as $x_2$. The objective is to maximize profit, which is $50x_1 + 35x_2$. The constraints are:
- The total number of rooms available is 40: $x_1 + x_2 \leq 40$
- At least 15 rooms must be heated: $x_1 \geq 15$
- At least 4 times as many people prefer unheated rooms to heated rooms: $x_2 \geq 4x_1$
- Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

However, since $x_1$ and $x_2$ represent the number of rooms, they must be integers, but for the purpose of linear programming, we'll first solve it as if $x_1$ and $x_2$ can be continuous.

## 2: Convert the problem into a symbolic representation
The symbolic representation can be written as:
```json
{
'sym_variables': [('x1', 'heated rooms'), ('x2', 'unheated rooms')],
'objective_function': '50*x1 + 35*x2',
'constraints': [
    'x1 + x2 <= 40',
    'x1 >= 15',
    'x2 >= 4*x1',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 3: Adjust the constraints for practical integer solutions
Given that $x_1 \geq 15$ and $x_2 \geq 4x_1$, substituting $x_1 = 15$ into $x_2 \geq 4x_1$ yields $x_2 \geq 60$. However, $x_1 + x_2 \leq 40$ cannot be satisfied if $x_2 \geq 60$ and $x_1 = 15$ because $15 + 60 > 40$. This indicates a potential issue with finding a feasible solution that satisfies all constraints, especially considering the constraint that at least 4 times as many people prefer unheated rooms to heated rooms.

## 4: Solve using Gurobi
To solve this problem using Gurobi, we need to correct and slightly adjust our approach to ensure we're maximizing profit under the given constraints.

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="heated_rooms", lb=0, vtype=gurobi.GRB.INTEGER)  # Heated rooms
    x2 = model.addVar(name="unheated_rooms", lb=0, vtype=gurobi.GRB.INTEGER)  # Unheated rooms

    # Objective function: Maximize profit
    model.setObjective(50 * x1 + 35 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(x1 + x2 <= 40, name="total_rooms")
    model.addConstr(x1 >= 15, name="min_heated")
    model.addConstr(x2 >= 4 * x1, name="unheated_preference")

    # Solve the model
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Heated rooms = {x1.varValue}, Unheated rooms = {x2.varValue}")
        print(f"Max Profit: ${50*x1.varValue + 35*x2.varValue}")
    else:
        print("No optimal solution found")

solve_hostel_problem()
```

Given the constraints and upon inspection, it seems there might be an infeasibility due to the constraints provided, especially with $x_2 \geq 4x_1$ and $x_1 + x_2 \leq 40$ combined with $x_1 \geq 15$. Let's verify:

If $x_1 = 15$, then $x_2 \geq 60$ which violates $x_1 + x_2 \leq 40$. 

The problem as stated may not have a feasible solution that satisfies all given constraints, indicating a potential need to revisit or relax some constraints. 

However, executing the Gurobi code will provide a clear answer based on the model's optimization attempt. 

```python
import gurobi

def solve_hostel_problem():
    model = gurobi.Model()

    x1 = model.addVar(name="heated_rooms", lb=0, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="unheated_rooms", lb=0, vtype=gurobi.GRB.INTEGER)

    model.setObjective(50 * x1 + 35 * x2, gurobi.GRB.MAXIMIZE)

    model.addConstr(x1 + x2 <= 40, name="total_rooms")
    model.addConstr(x1 >= 15, name="min_heated")
    model.addConstr(x2 >= 4 * x1, name="unheated_preference")

    model.optimize()

    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Heated rooms = {x1.varValue}, Unheated rooms = {x2.varValue}")
        print(f"Max Profit: ${50*x1.varValue + 35*x2.varValue}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible")
    else:
        print("No optimal solution found")

solve_hostel_problem()
```