## Symbolic Representation

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

- $x_1$ = Number of bags of family mix
- $x_2$ = Number of bags of party mix

The objective is to maximize profit. The profit from selling $x_1$ bags of family mix at $4.5 per pound and $x_2$ bags of party mix at $6 per pound needs to be calculated based on the weight of each type of popcorn used.

## Assumptions
- Assume a bag of family mix or party mix weighs 1 pound for simplicity.

## Objective Function

The family mix has 50% caramel and 50% butter popcorn. The party mix has 80% caramel and 20% butter popcorn.

Let's denote:
- $B$ = pounds of butter popcorn available = 35 pounds
- $C$ = pounds of caramel popcorn available = 45 pounds

The amount of butter popcorn used in $x_1$ family mixes and $x_2$ party mixes is $0.5x_1 + 0.2x_2$.
The amount of caramel popcorn used in $x_1$ family mixes and $x_2$ party mixes is $0.5x_1 + 0.8x_2$.

The profit $P$ from selling $x_1$ bags of family mix at $4.5 per pound and $x_2$ bags of party mix at $6 per pound is:
\[ P = 4.5x_1 + 6x_2 \]

## Constraints

1. Butter popcorn constraint: $0.5x_1 + 0.2x_2 \leq 35$
2. Caramel popcorn constraint: $0.5x_1 + 0.8x_2 \leq 45$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON

```json
{
    'sym_variables': [('x1', 'family mix'), ('x2', 'party mix')],
    'objective_function': '4.5*x1 + 6*x2',
    'constraints': [
        '0.5*x1 + 0.2*x2 <= 35',
        '0.5*x1 + 0.8*x2 <= 45',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code

```python
import gurobi

def solve_popcorn_problem():
    # Create a new model
    m = gurobi.Model()

    # Define variables
    x1 = m.addVar(name="family_mix", lb=0, vtype=gurobi.GRB.CONTINUOUS)
    x2 = m.addVar(name="party_mix", lb=0, vtype=gurobi.GRB.CONTINUOUS)

    # Objective function
    m.setObjective(4.5 * x1 + 6 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    m.addConstr(0.5 * x1 + 0.2 * x2 <= 35, name="butter_constraint")
    m.addConstr(0.5 * x1 + 0.8 * x2 <= 45, name="caramel_constraint")

    # Optimize
    m.optimize()

    # Print solution
    if m.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution:")
        print(f"Family mix: {x1.varValue} bags")
        print(f"Party mix: {x2.varValue} bags")
        print(f"Max profit: {m.objVal}")
    else:
        print("No optimal solution found")

solve_popcorn_problem()
```