## Problem Description and Formulation

The young entrepreneur has $500,000 to invest in two sectors: healthcare and energy. The goal is to maximize the return on investment (ROI) over a 12-month period. The healthcare sector yields a 4% return, while the energy sector yields an 8% return. There are two constraints:

1. At least 60% of the investment must be in the healthcare sector.
2. At most 35% of the investment can be in the energy sector.

## Mathematical Formulation

Let \(H\) be the amount invested in the healthcare sector and \(E\) be the amount invested in the energy sector. The objective is to maximize the total return on investment, which can be represented as \(0.04H + 0.08E\).

The constraints are:

1. \(H + E \leq 500,000\) (total investment not exceeding $500,000)
2. \(H \geq 0.6 \times 500,000\) (at least 60% in healthcare)
3. \(E \leq 0.35 \times 500,000\) (at most 35% in energy)
4. \(H, E \geq 0\) (non-negativity constraint, investments cannot be negative)

## Gurobi Code

```python
import gurobi

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

    # Define variables
    H = model.addVar(name="Healthcare", lb=0)  # Investment in healthcare
    E = model.addVar(name="Energy", lb=0)     # Investment in energy

    # Objective: Maximize return on investment
    model.setObjective(0.04 * H + 0.08 * E, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(H + E <= 500000, name="Total_Investment")
    model.addConstr(H >= 0.6 * 500000, name="Healthcare_Min")
    model.addConstr(E <= 0.35 * 500000, name="Energy_Max")

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print(f"Optimal investment in Healthcare: ${H.varValue:.2f}")
        print(f"Optimal investment in Energy: ${E.varValue:.2f}")
        print(f"Max Return: ${0.04 * H.varValue + 0.08 * E.varValue:.2f}")
    else:
        print("The model is infeasible.")

solve_investment_problem()
```