To solve this optimization problem, we will use linear programming. The goal is to maximize profit given the constraints on silicon, design time, soldering time, and production minimums for small and large PCBs.

Let's denote:
- \(x_s\) as the number of small PCBs produced,
- \(x_l\) as the number of large PCBs produced.

The objective function (to be maximized) is the total profit from producing both types of PCBs:
\[ \text{Maximize:} \quad 20x_s + 35x_l \]

Given constraints are:
1. Silicon availability: \(3x_s + 5x_l \leq 250\)
2. Design time availability: \(30x_s + 40x_l \leq 800\)
3. Soldering time availability: \(20x_s + 30x_l \leq 600\)
4. Minimum production of small PCBs: \(x_s \geq 5\)
5. Minimum production of large PCBs: \(x_l \geq 6\)

All variables are non-negative since they represent quantities of products.

Now, let's translate this into Gurobi code in Python:

```python
from gurobipy import *

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

# Create variables
x_s = m.addVar(lb=5, vtype=GRB.INTEGER, name="small_pcb")
x_l = m.addVar(lb=6, vtype=GRB.INTEGER, name="large_pcb")

# Objective function: Maximize profit
m.setObjective(20*x_s + 35*x_l, GRB.MAXIMIZE)

# Constraints
m.addConstr(3*x_s + 5*x_l <= 250, "silicon")
m.addConstr(30*x_s + 40*x_l <= 800, "design_time")
m.addConstr(20*x_s + 30*x_l <= 600, "soldering_time")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Small PCBs: {x_s.x}")
    print(f"Large PCBs: {x_l.x}")
    print(f"Total Profit: ${20*x_s.x + 35*x_l.x:.2f}")
else:
    print("No optimal solution found")
```