## Step 1: Define the symbolic representation of the problem

Let's denote:
- $x_1$ as the number of vehicle tickets sold
- $x_2$ as the number of passenger tickets sold

The objective is to maximize profit. Given that the profit is $50 per vehicle ticket and $50 per passenger ticket, the objective function can be represented as:
\[ \text{Maximize:} \quad 50x_1 + 50x_2 \]

## Step 2: Identify the constraints

From the problem description, we have the following constraints:
1. The ferry can sell at most 100 tickets: $x_1 + x_2 \leq 100$
2. A minimum of 10 tickets for vehicles: $x_1 \geq 10$
3. At least 5 times as many people buy passenger tickets than vehicle tickets: $x_2 \geq 5x_1$

## 3: Symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'vehicle tickets'), ('x2', 'passenger tickets')],
    'objective_function': 'Maximize: 50x1 + 50x2',
    'constraints': [
        'x1 + x2 <= 100',
        'x1 >= 10',
        'x2 >= 5x1'
    ]
}
```

## Step 4: Convert the problem into Gurobi code

To solve this linear programming problem using Gurobi in Python, we will use the following code:

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="vehicle_tickets", lb=0, ub=None, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="passenger_tickets", lb=0, ub=None, vtype=gurobi.GRB.INTEGER)

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

    # Constraints
    model.addConstr(x1 + x2 <= 100, name="total_tickets")
    model.addConstr(x1 >= 10, name="min_vehicle_tickets")
    model.addConstr(x2 >= 5*x1, name="passenger_to_vehicle_ratio")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Vehicle tickets = {x1.varValue}, Passenger tickets = {x2.varValue}")
    else:
        print("The model is infeasible")

solve_ferry_ticket_problem()
```