## Symbolic Representation of the Optimization Problem

The optimization problem can be symbolically represented as follows:

Let's denote:
- $x_1$ as the number of sports ads
- $x_2$ as the number of TV show commercials

The objective is to minimize the total cost, which can be represented as:
- $90,000x_1 + 20,000x_2$

The constraints are:
- $4x_1 + 12x_2 \geq 40$ (to reach at least 40 million baby boomers)
- $18x_1 + 5x_2 \geq 25$ (to reach at least 25 million millennials)
- $x_1, x_2 \geq 0$ (non-negativity constraints, as the number of ads cannot be negative)

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'sports ads'), ('x2', 'TV show commercials')],
    'objective_function': '90000*x1 + 20000*x2',
    'constraints': [
        '4*x1 + 12*x2 >= 40',
        '18*x1 + 5*x2 >= 25',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define the variables
x1 = model.addVar(name="sports_ads", lb=0, vtype=gp.GRB.INTEGER)  # Number of sports ads
x2 = model.addVar(name="tv_show_commercials", lb=0, vtype=gp.GRB.INTEGER)  # Number of TV show commercials

# Define the objective function
model.setObjective(90000 * x1 + 20000 * x2, gp.GRB.MINIMIZE)

# Add constraints
model.addConstr(4 * x1 + 12 * x2 >= 40, name="baby_boomers_constraint")  # Baby boomers constraint
model.addConstr(18 * x1 + 5 * x2 >= 25, name="millennials_constraint")  # Millennials constraint

# Solve the model
model.optimize()

# Print the solution
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution found.")
    print(f"Number of sports ads: {x1.varValue}")
    print(f"Number of TV show commercials: {x2.varValue}")
    print(f"Minimum cost: ${model.objVal:.2f}")
else:
    print("No optimal solution found.")
```