## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of bookcases
- $x_2$ represents the number of dining tables

## Step 2: Formulate the objective function
The profit per bookcase is $150 and the profit per dining table is $200. The objective function to maximize profit is:
\[ \text{Maximize:} \quad 150x_1 + 200x_2 \]

## 3: Define the constraints
1. Floor space constraint: $15x_1 + 8x_2 \leq 1200$
2. Bookcase percentage constraint: $x_1 \geq 0.20(x_1 + x_2)$
3. Capital constraint: $1200x_1 + 1500x_2 \leq 50000$
4. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## 4: Convert the bookcase percentage constraint to a standard form
The bookcase percentage constraint can be rewritten as:
\[ x_1 \geq 0.20x_1 + 0.20x_2 \]
\[ 0.80x_1 \geq 0.20x_2 \]
\[ 4x_1 \geq x_2 \]
\[ 4x_1 - x_2 \geq 0 \]

## 5: Symbolic representation of the problem
```json
{
'sym_variables': [('x1', 'bookcases'), ('x2', 'dining tables')],
'objective_function': '150*x1 + 200*x2',
'constraints': [
    '15*x1 + 8*x2 <= 1200',
    '4*x1 - x2 >= 0',
    '1200*x1 + 1500*x2 <= 50000',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 6: Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="bookcases", lb=0)
    x2 = model.addVar(name="dining_tables", lb=0)

    # Objective function
    model.setObjective(150 * x1 + 200 * x2, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(15 * x1 + 8 * x2 <= 1200, name="floor_space")
    model.addConstr(4 * x1 - x2 >= 0, name="bookcase_percentage")
    model.addConstr(1200 * x1 + 1500 * x2 <= 50000, name="capital")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of bookcases: {x1.varValue}")
        print(f"Number of dining tables: {x2.varValue}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_optimization_problem()
```