## Problem Description and Formulation

The problem is a linear programming optimization problem. The goal is to maximize the profit of producing 1st and 2nd generation motherboards given certain constraints on resources (silicon, assembly time, and soldering time) and minimum production requirements.

### Variables
- \(x_1\): Number of 1st generation motherboards to produce.
- \(x_2\): Number of 2nd generation motherboards to produce.

### Objective Function
Maximize profit \(P = 100x_1 + 125x_2\).

### Constraints
1. Silicon constraint: \(4x_1 + 6x_2 \leq 450\)
2. Assembly time constraint: \(20x_1 + 30x_2 \leq 900\)
3. Soldering time constraint: \(30x_1 + 40x_2 \leq 500\)
4. Minimum production constraints:
   - \(x_1 \geq 6\)
   - \(x_2 \geq 7\)
5. Non-negativity constraints:
   - \(x_1 \geq 0\)
   - \(x_2 \geq 0\)

Since \(x_1\) and \(x_2\) represent quantities of motherboards, they must be integers.

## Gurobi Code

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(lb=6, vtype=gurobi.GRB.INTEGER, name="1st_generation_motherboards")
    x2 = model.addVar(lb=7, vtype=gurobi.GRB.INTEGER, name="2nd_generation_motherboards")

    # Objective function: Maximize profit
    model.setObjective(100*x1 + 125*x2, gurobi.GRB.MAXIMIZE)

    # Silicon constraint
    model.addConstr(4*x1 + 6*x2 <= 450, name="silicon_constraint")

    # Assembly time constraint
    model.addConstr(20*x1 + 30*x2 <= 900, name="assembly_time_constraint")

    # Soldering time constraint
    model.addConstr(30*x1 + 40*x2 <= 500, name="soldering_time_constraint")

    # Solve the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal solution: Produce {x1.varValue} 1st generation motherboards and {x2.varValue} 2nd generation motherboards.")
        print(f"Maximum profit: ${model.objVal:.2f}")
    else:
        print("No optimal solution found.")

solve_motherboard_production()
```