To solve the optimization problem described, we first need to convert the natural language description into a symbolic representation. This involves defining variables, an objective function, and constraints based on the given information.

Let's denote:
- $x_1$ as the number of tennis rackets to be made,
- $x_2$ as the number of badminton rackets to be made.

The objective is to maximize profit. Given that each tennis racket yields a profit of $20 and each badminton racket yields a profit of $17, the objective function can be written as:
\[ \text{Maximize} \quad 20x_1 + 17x_2 \]

The constraints are based on the time available for molding and stringing. For molding, there are 3000 minutes available, with each tennis racket requiring 12 minutes and each badminton racket requiring 10 minutes:
\[ 12x_1 + 10x_2 \leq 3000 \]

For stringing, there are 3500 minutes available, with each tennis racket requiring 15 minutes and each badminton racket requiring 12 minutes:
\[ 15x_1 + 12x_2 \leq 3500 \]

Additionally, the number of rackets cannot be negative, so we have non-negativity constraints:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Now, let's represent this problem symbolically and in JSON format as requested:

```json
{
    'sym_variables': [('x1', 'number of tennis rackets'), ('x2', 'number of badminton rackets')],
    'objective_function': '20*x1 + 17*x2',
    'constraints': ['12*x1 + 10*x2 <= 3000', '15*x1 + 12*x2 <= 3500', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="tennis_rackets")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="badminton_rackets")

# Set the objective function
m.setObjective(20*x1 + 17*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(12*x1 + 10*x2 <= 3000, "molding_time")
m.addConstr(15*x1 + 12*x2 <= 3500, "stringing_time")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of tennis rackets: {x1.x}")
    print(f"Number of badminton rackets: {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```