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

- $x_1$ represents the number of modern violins made.
- $x_2$ represents the number of baroque violins made.

The objective function, which is to maximize profits, can be represented algebraically as:
\[ 150x_1 + 200x_2 \]

Given that a modern violin takes 5 hours of woodworking time and a baroque violin takes 4 hours, and there are 150 hours available per day for woodworking among all the workers, we have our first constraint:
\[ 5x_1 + 4x_2 \leq 150 \]

For assembling time, with modern violins requiring 3.5 hours and baroque violins requiring 5 hours, and there being 200 hours available per day for assembling among all the workers, we have our second constraint:
\[ 3.5x_1 + 5x_2 \leq 200 \]

Additionally, since we cannot produce a negative number of violins, we have non-negativity constraints:
\[ x_1 \geq 0 \]
\[ x_2 \geq 0 \]

Thus, the symbolic representation in JSON format is:

```json
{
    'sym_variables': [('x1', 'number of modern violins'), ('x2', 'number of baroque violins')],
    'objective_function': '150*x1 + 200*x2',
    'constraints': ['5*x1 + 4*x2 <= 150', '3.5*x1 + 5*x2 <= 200', 'x1 >= 0', 'x2 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:

```python
from gurobipy import *

# Create a new model
m = Model("Violin_Factory")

# Define variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="modern_violins", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="baroque_violins", lb=0)

# Set objective function
m.setObjective(150*x1 + 200*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(5*x1 + 4*x2 <= 150, "woodworking_time")
m.addConstr(3.5*x1 + 5*x2 <= 200, "assembling_time")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print(f"Optimal solution: {x1.varName} = {x1.x}, {x2.varName} = {x2.x}")
    print(f"Maximum profit: {m.objVal}")
else:
    print("No optimal solution found")
```