## Symbolic Representation

To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's denote:

- \(x_1\) as the number of toques
- \(x_2\) as the number of scarfs

The objective is to maximize profit. Given that the profit per toque is $5 and per scarf is $7, the objective function can be written as:

\[ \text{Maximize:} \quad 5x_1 + 7x_2 \]

The constraints based on the given resources are:

1. Yarn constraint: \(3x_1 + 5x_2 \leq 200\)
2. Time constraint: \(30x_1 + 40x_2 \leq 1800\)
3. Non-negativity constraints: \(x_1 \geq 0, x_2 \geq 0\)

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'toques'), ('x2', 'scarfs')],
    'objective_function': '5*x1 + 7*x2',
    'constraints': [
        '3*x1 + 5*x2 <= 200',
        '30*x1 + 40*x2 <= 1800',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

# Create a new model
model = gp.Model("Knitting_Problem")

# Define variables
x1 = model.addVar(name="toques", lb=0, vtype=gp.GRB.INTEGER)  # Number of toques
x2 = model.addVar(name="scarfs", lb=0, vtype=gp.GRB.INTEGER)  # Number of scarfs

# Objective function: Maximize profit
model.setObjective(5 * x1 + 7 * x2, gp.GRB.MAXIMIZE)

# Constraints
model.addConstr(3 * x1 + 5 * x2 <= 200, name="yarn_constraint")  # Yarn constraint
model.addConstr(30 * x1 + 40 * x2 <= 1800, name="time_constraint")  # Time constraint

# Solve the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution: Toques = {x1.varValue}, Scarfs = {x2.varValue}")
    print(f"Maximum profit: ${5 * x1.varValue + 7 * x2.varValue}")
else:
    print("No optimal solution found.")
```