To solve this optimization problem, we first need to convert the natural language description into a symbolic representation. Let's denote the number of shuttlecocks as \(x_1\) and the number of volleyballs as \(x_2\).

The objective is to maximize profit. The profit per shuttlecock is $3.5, and the profit per volleyball is $10. Thus, the objective function can be represented algebraically as:
\[ \text{Maximize:} \quad 3.5x_1 + 10x_2 \]

Now, let's consider the constraints:

1. **Sewing Time Constraint**: To make one shuttlecock requires 15 minutes of sewing, and to make a volleyball requires 20 minutes of sewing. The total available sewing time in a month is 4000 minutes. Therefore, the constraint can be represented as:
\[ 15x_1 + 20x_2 \leq 4000 \]

2. **Quality Checking Time Constraint**: To make one shuttlecock requires 5 minutes of quality checking, and to make a volleyball requires 10 minutes of quality checking. The total available quality checking time in a month is 3000 minutes. Therefore, the constraint can be represented as:
\[ 5x_1 + 10x_2 \leq 3000 \]

3. **Non-Negativity Constraints**: Since we cannot produce a negative number of shuttlecocks or volleyballs, both \(x_1\) and \(x_2\) must be non-negative:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of shuttlecocks'), ('x2', 'number of volleyballs')],
  'objective_function': '3.5*x1 + 10*x2',
  'constraints': ['15*x1 + 20*x2 <= 4000', '5*x1 + 10*x2 <= 3000', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can write the following code:
```python
from gurobipy import *

# Create a new model
m = Model("Sports_Company_Optimization")

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

# Set the objective function
m.setObjective(3.5*x1 + 10*x2, GRB.MAXIMIZE)

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

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of shuttlecocks: {x1.x}")
    print(f"Number of volleyballs: {x2.x}")
    print(f"Maximum profit: ${3.5*x1.x + 10*x2.x:.2f}")
else:
    print("No optimal solution found")
```