To solve the given problem, we first need to convert the natural language description into a symbolic representation. Let's define:

- $x_1$ as the number of commercials purchased during cartoons.
- $x_2$ as the number of commercials purchased during kids-movies.

The objective is to minimize the total cost. The cost for a commercial during a cartoon is $5000, and the cost for a commercial during a kids-movie is $12000. Therefore, the objective function can be represented as:

\[ \text{Minimize} \quad 5000x_1 + 12000x_2 \]

The constraints are based on the requirements that the commercials should be seen by at least 30 million young boys and 40 million young girls. Given that each cartoon is seen by 2 million young boys and 1 million young girls, and each kids-movie is seen by 4 million young boys and 6 million young girls, we have:

- For young boys: $2x_1 + 4x_2 \geq 30$
- For young girls: $x_1 + 6x_2 \geq 40$

Additionally, since the number of commercials cannot be negative, we also have:

- $x_1 \geq 0$
- $x_2 \geq 0$

Therefore, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'number of commercials during cartoons'), ('x2', 'number of commercials during kids-movies')],
    'objective_function': '5000*x1 + 12000*x2',
    'constraints': ['2*x1 + 4*x2 >= 30', 'x1 + 6*x2 >= 40', 'x1 >= 0', 'x2 >= 0']
}
```

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

```python
from gurobipy import *

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

# Define the variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="cartoon_commercials", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="kids_movie_commercials", lb=0)

# Set the objective function
m.setObjective(5000*x1 + 12000*x2, GRB.MINIMIZE)

# Add constraints
m.addConstr(2*x1 + 4*x2 >= 30, "young_boys_constraint")
m.addConstr(x1 + 6*x2 >= 40, "young_girls_constraint")

# Optimize the model
m.optimize()

# Print the results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Number of commercials during cartoons: {x1.x}")
    print(f"Number of commercials during kids-movies: {x2.x}")
    print(f"Total cost: {m.objVal}")
else:
    print("No optimal solution found")
```