To solve this optimization problem, we first need to define the decision variables and the objective function. Let's denote the number of lumbers produced as \(L\) and the number of plywoods produced as \(P\). The objective is to maximize profit, which is $10 per lumber and $35 per plywood. So, the total profit can be represented as \(10L + 35P\).

The constraints are based on the available worker-hours and machine-hours. For worker-hours, each lumber requires 3 hours and each plywood requires 2 hours, with a total of 2500 hours available. This gives us the constraint \(3L + 2P \leq 2500\). For machine-hours, each lumber requires 8 hours and each plywood requires 12 hours, with a total of 4000 hours available, leading to the constraint \(8L + 12P \leq 4000\).

Additionally, there are minimum production requirements: at least 200 lumbers (\(L \geq 200\)) and at least 300 plywoods (\(P \geq 300\)).

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

```python
from gurobipy import *

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

# Define the decision variables
L = m.addVar(vtype=GRB.INTEGER, name="Lumbers", lb=200)  # At least 200 lumbers
P = m.addVar(vtype=GRB.INTEGER, name="Plywoods", lb=300)  # At least 300 plywoods

# Set the objective function: Maximize profit
m.setObjective(10*L + 35*P, GRB.MAXIMIZE)

# Add constraints
# Worker-hours constraint
m.addConstr(3*L + 2*P <= 2500, name="Worker_Hours")

# Machine-hours constraint
m.addConstr(8*L + 12*P <= 4000, name="Machine_Hours")

# Optimize the model
m.optimize()

# Print results
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Lumbers to produce: {L.x}")
    print(f"Plywoods to produce: {P.x}")
    print(f"Maximum profit: ${10*L.x + 35*P.x}")
else:
    print("No optimal solution found. The model is infeasible.")

```

This code defines the optimization problem, sets up the decision variables with their lower bounds (to ensure the minimum production requirements are met), specifies the objective function to maximize profit, and adds the constraints related to worker-hours and machine-hours. It then solves the model and prints out the optimal number of lumbers and plywoods to produce along with the maximum achievable profit. If no feasible solution exists, it indicates that the problem is infeasible.