## Symbolic Representation

To solve the given optimization problem, we first need to convert the natural language description into a symbolic representation.

Let's define the symbolic variables as follows:

- $x_1$ represents the number of A400 keyboards produced.
- $x_2$ represents the number of P500 keyboards produced.

The objective is to maximize profit. Given that one A400 keyboard yields a $35 profit and one P500 keyboard yields an $80 profit, the objective function can be represented as:

Maximize $35x_1 + 80x_2$

The constraints given are:

1. The business can spend up to 45 hours a week to manufacture these keyboards. Since making one A400 keyboard requires 5 hours of labour and one P500 keyboard requires 9 hours of labour, we have:
   $5x_1 + 9x_2 \leq 45$

2. The business decides to produce at least three times as many A400 keyboards as P500 ones:
   $x_1 \geq 3x_2$

3. Non-negativity constraints, as the number of keyboards cannot be negative:
   $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
'sym_variables': [('x1', 'A400 keyboards'), ('x2', 'P500 keyboards')],
'objective_function': '35*x1 + 80*x2',
'constraints': [
    '5*x1 + 9*x2 <= 45',
    'x1 >= 3*x2',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## Gurobi Code in Python

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="A400 keyboards", lb=0, vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name="P500 keyboards", lb=0, vtype=gurobi.GRB.INTEGER)

    # Objective function: Maximize 35*x1 + 80*x2
    model.setObjective(35*x1 + 80*x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(5*x1 + 9*x2 <= 45, name="labour_hours")
    model.addConstr(x1 >= 3*x2, name="production_ratio")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal production levels: A400 = {x1.varValue}, P500 = {x2.varValue}")
        print(f"Maximum profit: ${model.objVal:.2f}")
    else:
        print("The problem is infeasible.")

solve_optimization_problem()
```