## Problem Description and Formulation

The problem described is a classic example of a linear programming problem. The goal is to maximize profit given certain constraints on resources (yarn and time) and the production of two items: toques and scarfs.

### Variables
- Let \(T\) be the number of toques produced.
- Let \(S\) be the number of scarfs produced.

### Objective Function
The profit per toque is $5, and the profit per scarf is $7. The objective is to maximize the total profit \(P\), which can be represented as:
\[ P = 5T + 7S \]

### Constraints
1. **Yarn Constraint**: A toque requires 3 units of yarn, and a scarf requires 5 units of yarn. The man has 200 units of yarn available. This can be represented as:
\[ 3T + 5S \leq 200 \]

2. **Time Constraint**: A toque requires 30 minutes of time, and a scarf requires 40 minutes of time. The man has 1800 minutes of time available. This can be represented as:
\[ 30T + 40S \leq 1800 \]

3. **Non-Negativity Constraint**: The number of toques and scarfs produced cannot be negative:
\[ T \geq 0, S \geq 0 \]

### Gurobi Code

To solve this problem using Gurobi in Python, we will first install the Gurobi library if it's not already installed. You can install it using pip:
```bash
pip install gurobi
```

Here is the Gurobi code for the problem:

```python
import gurobi as gp

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

# Define variables
T = model.addVar(name="Toques", lb=0, vtype=gp.GRB.INTEGER)  # Number of toques
S = model.addVar(name="Scarfs", lb=0, vtype=gp.GRB.INTEGER)  # Number of scarfs

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

# Constraints
model.addConstr(3 * T + 5 * S <= 200, name="Yarn_Constraint")  # Yarn constraint
model.addConstr(30 * T + 40 * S <= 1800, name="Time_Constraint")  # Time constraint

# Solve the model
model.optimize()

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