To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for the quantities of toques and scarves, formulating the objective function that represents the total profit, and listing the constraints based on the available resources (yarn and time).

Let's define:
- $x_1$ as the number of toques knitted
- $x_2$ as the number of scarves knitted

The objective function aims to maximize the total profit. Given that each toque yields a profit of $5 and each scarf yields a profit of $7, the objective function can be represented algebraically as:
\[ \text{Maximize:} \quad 5x_1 + 7x_2 \]

Now, let's consider the constraints:
1. **Yarn Constraint:** Each toque requires 3 units of yarn, and each scarf requires 5 units of yarn. The total available yarn is 200 units. Therefore, the constraint can be represented as:
\[ 3x_1 + 5x_2 \leq 200 \]
2. **Time Constraint:** Knitting a toque takes 30 minutes, and knitting a scarf takes 40 minutes. The total available time is 1800 minutes. Hence, the time constraint is:
\[ 30x_1 + 40x_2 \leq 1800 \]
3. **Non-Negativity Constraints:** Since the number of toques and scarves cannot be negative, we have:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

In symbolic notation with natural language objects, our problem can be summarized as:

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

Now, let's implement this problem using Gurobi in Python:

```python
from gurobipy import *

# Create a new model
model = Model("Knitting_Optimization")

# Define the variables
x1 = model.addVar(vtype=GRB.INTEGER, name="toques")
x2 = model.addVar(vtype=GRB.INTEGER, name="scarves")

# Set the objective function: Maximize profit
model.setObjective(5*x1 + 7*x2, GRB.MAXIMIZE)

# Add constraints
model.addConstr(3*x1 + 5*x2 <= 200, "yarn_constraint")
model.addConstr(30*x1 + 40*x2 <= 1800, "time_constraint")

# Optimize the model
model.optimize()

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