## Step 1: Define the symbolic representation of the problem
Let's denote the number of mini bears as $x_1$ and the number of mini dogs as $x_2$. The objective is to maximize profit, which is $40x_1 + 47x_2$. The constraints are:
- $8x_1 + 7x_2 \leq 400$ (cotton constraint)
- $x_1 + x_2 \leq 40$ (time constraint)
- $x_1 \geq 0$ and $x_2 \geq 0$ (non-negativity constraints)

## Step 2: Convert the problem into a symbolic representation
The symbolic representation can be written as:
```json
{
    'sym_variables': [('x1', 'mini bears'), ('x2', 'mini dogs')],
    'objective_function': '40*x1 + 47*x2',
    'constraints': [
        '8*x1 + 7*x2 <= 400',
        'x1 + x2 <= 40',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 3: Implement the problem in Gurobi code
To solve this linear programming problem using Gurobi, we will use the Gurobi Python API.

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="mini_bears", lb=0, ub=None, obj=40)
    x2 = model.addVar(name="mini_dogs", lb=0, ub=None, obj=47)

    # Add constraints
    model.addConstr(x1 + x2 <= 40, name="time_constraint")
    model.addConstr(8*x1 + 7*x2 <= 400, name="cotton_constraint")

    # Set the model objective
    model.setObjective(x1.obj * x1.varName + x2.obj * x2.varName, gurobi.GRB.MAXIMIZE)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Mini bears: {x1.x}")
        print(f"Mini dogs: {x2.x}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```