Here's the formulation of the linear program and the Gurobi code to solve it:

**Decision Variables:**

* `x`: Number of small PCBs produced.
* `y`: Number of large PCBs produced.

**Objective Function:**

Maximize profit: `20x + 35y`

**Constraints:**

* **Silicon:** `3x + 5y <= 250`
* **Design Time:** `30x + 40y <= 800` (minutes)
* **Soldering Time:** `20x + 30y <= 600` (minutes)
* **Minimum Small PCBs:** `x >= 5`
* **Minimum Large PCBs:** `y >= 6`
* **Non-negativity:** `x, y >= 0` (implicitly handled by Gurobi for continuous variables)


```python
import gurobipy as gp

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

# Create decision variables
x = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="small_PCBs") #allowing fractional PCBs for now
y = model.addVar(lb=0, vtype=gp.GRB.CONTINUOUS, name="large_PCBs")

# Set objective function
model.setObjective(20*x + 35*y, gp.GRB.MAXIMIZE)

# Add constraints
model.addConstr(3*x + 5*y <= 250, "silicon_constraint")
model.addConstr(30*x + 40*y <= 800, "design_constraint")
model.addConstr(20*x + 30*y <= 600, "soldering_constraint")
model.addConstr(x >= 5, "min_small_PCBs")
model.addConstr(y >= 6, "min_large_PCBs")


# Optimize the model
model.optimize()

# Print the results
if model.status == gp.GRB.OPTIMAL:
    print(f"Optimal Solution Found:")
    print(f"Number of small PCBs: {x.x}")
    print(f"Number of large PCBs: {y.x}")
    print(f"Maximum Profit: ${model.objVal}")
elif model.status == gp.GRB.INFEASIBLE:
    print("The model is infeasible.")
else:
    print(f"Optimization terminated with status: {model.status}")

```
