## Problem Description and Symbolic Representation

The problem involves maximizing profit by determining the number of small and large plush toys to produce given certain constraints on cotton and worker time.

### Symbolic Variables:
- $x_1$ = Number of small plush toys
- $x_2$ = Number of large plush toys

### Objective Function:
Maximize profit $P = 3x_1 + 5x_2$

### Constraints:
1. Cotton constraint: $5x_1 + 8x_2 \leq 250$
2. Worker time constraint: $10x_1 + 12x_2 \leq 500$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## Symbolic Representation in JSON Format

```json
{
    'sym_variables': [('x1', 'small plush toys'), ('x2', 'large plush toys')],
    'objective_function': '3*x1 + 5*x2',
    'constraints': [
        '5*x1 + 8*x2 <= 250',
        '10*x1 + 12*x2 <= 500',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Gurobi Code in Python

```python
import gurobipy as gp

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

# Define variables
x1 = m.addVar(name="small_plush_toys", lb=0, vtype=gp.GRB.INTEGER)  # Number of small plush toys
x2 = m.addVar(name="large_plush_toys", lb=0, vtype=gp.GRB.INTEGER)  # Number of large plush toys

# Objective function: Maximize profit
m.setObjective(3*x1 + 5*x2, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(5*x1 + 8*x2 <= 250, name="cotton_constraint")  # Cotton constraint
m.addConstr(10*x1 + 12*x2 <= 500, name="worker_time_constraint")  # Worker time constraint

# Solve the model
m.optimize()

# Print the results
if m.status == gp.GRB.OPTIMAL:
    print(f"Optimal solution found. Produce {x1.varValue} small plush toys and {x2.varValue} large plush toys.")
    print(f"Maximum profit: ${3*x1.varValue + 5*x2.varValue}")
else:
    print("No optimal solution found.")
```