1. **State the problem:** Donneth wants to invest a total of 100000 in life insurance and a bank account. 2. **Define variables:** Let $x$ be the amount invested in life insurance and $y$ be the amount invested in the bank. 3. **Write constraints:** - Total investment: $$x + y = 100000$$ - Bank investment limit: $$y \leq 40000$$ - Life insurance minimum: $$x \geq 30000$$ 4. **Objective function:** Maximize returns $$R = 0.10x + 0.08y$$ 5. **Express $y$ in terms of $x$:** From total investment, $$y = 100000 - x$$ 6. **Rewrite returns:** $$R = 0.10x + 0.08(100000 - x) = 0.10x + 8000 - 0.08x = 0.02x + 8000$$ 7. **Analyze constraints on $x$:** - Since $y \leq 40000$, then $$100000 - x \leq 40000 \implies x \geq 60000$$ - Also, $x \geq 30000$ from life insurance minimum. So combined constraints on $x$ are: $$x \geq 60000$$ 8. **Maximize $R$:** Since $R = 0.02x + 8000$ increases as $x$ increases, maximize $x$ subject to $x \leq 100000$ (total investment). 9. **Choose $x = 100000$ (max possible):** Then, $$y = 100000 - 100000 = 0$$ 10. **Check constraints:** - $y = 0 \leq 40000$ (valid) - $x = 100000 \geq 30000$ (valid) 11. **Calculate maximum returns:** $$R = 0.10 \times 100000 + 0.08 \times 0 = 10000$$ **Final answer:** Donneth should invest 100000 in life insurance and 0 in the bank to maximize returns of 10000.