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

Let's denote the number of shuttlecocks as $x_1$ and the number of volleyballs as $x_2$. The profit per shuttlecock is $3.5, and the profit per volleyball is $10. The objective is to maximize the total profit.

## Step 2: Formulate the objective function

The objective function can be represented as $3.5x_1 + 10x_2$.

## Step 3: Define the constraints

To make one shuttlecock, 15 minutes of sewing and 5 minutes of quality checking are required. To make a volleyball, 20 minutes of sewing and 10 minutes of quality checking are required. In a month, 4000 minutes of sewing time and 3000 minutes of quality checking time are available. This translates to the following constraints:

- $15x_1 + 20x_2 \leq 4000$ (sewing time constraint)
- $5x_1 + 10x_2 \leq 3000$ (quality checking time constraint)
- $x_1 \geq 0$ and $x_2 \geq 0$ (non-negativity constraints)

## 4: Symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'shuttlecocks'), ('x2', 'volleyballs')],
    'objective_function': '3.5*x1 + 10*x2',
    'constraints': [
        '15*x1 + 20*x2 <= 4000',
        '5*x1 + 10*x2 <= 3000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 5: Gurobi code in Python

```python
import gurobipy as gp

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

# Define the variables
x1 = model.addVar(name="shuttlecocks", lb=0)
x2 = model.addVar(name="volleyballs", lb=0)

# Define the objective function
model.setObjective(3.5 * x1 + 10 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
model.addConstr(15 * x1 + 20 * x2 <= 4000, name="sewing_time")
model.addConstr(5 * x1 + 10 * x2 <= 3000, name="quality_checking_time")

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of shuttlecocks: {x1.varValue}")
    print(f"Number of volleyballs: {x2.varValue}")
    print(f"Max profit: {model.objVal}")
else:
    print("No optimal solution found.")
```