## Step 1: Define the symbolic representation of the problem

Let's define the symbolic variables:
- $x_1$ represents the number of bags of sweet mix
- $x_2$ represents the number of bags of regular mix

The objective is to maximize profit. The profit from selling $x_1$ bags of sweet mix at $3 per pound and $x_2$ bags of regular mix at $2 per pound needs to be calculated based on the weight of each mix. Assuming a standard bag weight (let's say 1 pound for simplicity, as the actual weight isn't specified), the profit can be directly related to the number of bags sold.

## Step 2: Determine the objective function

The sweet mix has 75% caramel and 25% butter popcorn. The regular mix has 50% caramel and 50% butter popcorn. 

Given:
- 40 pounds of caramel popcorn
- 30 pounds of butter popcorn

The amount of caramel popcorn used in $x_1$ bags of sweet mix and $x_2$ bags of regular mix is $0.75x_1 + 0.5x_2$.
The amount of butter popcorn used in $x_1$ bags of sweet mix and $x_2$ bags of regular mix is $0.25x_1 + 0.5x_2$.

## 3: Formulate the constraints

Constraints based on available popcorn:
1. $0.75x_1 + 0.5x_2 \leq 40$ (caramel popcorn constraint)
2. $0.25x_1 + 0.5x_2 \leq 30$ (butter popcorn constraint)

Non-negativity constraints:
3. $x_1 \geq 0$
4. $x_2 \geq 0$

## 4: Define the objective function algebraically

The profit $P$ from selling $x_1$ bags of sweet mix at $3 per pound and $x_2$ bags of regular mix at $2 per pound is:
$P = 3x_1 + 2x_2$

## 5: Symbolic representation

```json
{
    'sym_variables': [('x1', 'bags of sweet mix'), ('x2', 'bags of regular mix')],
    'objective_function': '3*x1 + 2*x2',
    'constraints': [
        '0.75*x1 + 0.5*x2 <= 40',
        '0.25*x1 + 0.5*x2 <= 30',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## 6: Gurobi code

```python
import gurobipy as gp

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

# Define variables
x1 = m.addVar(name="sweet_mix", lb=0)  # bags of sweet mix
x2 = m.addVar(name="regular_mix", lb=0)  # bags of regular mix

# Objective function: maximize profit
m.setObjective(3*x1 + 2*x2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(0.75*x1 + 0.5*x2 <= 40, name="caramel_constraint")
m.addConstr(0.25*x1 + 0.5*x2 <= 30, name="butter_constraint")

# Solve the model
m.solve()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution:")
    print(f"Bags of sweet mix: {x1.varValue}")
    print(f"Bags of regular mix: {x2.varValue}")
    print(f"Max profit: {m.objVal}")
else:
    print("No optimal solution found")
```