1. **State the problem:** You have 100 lumber to allocate between buying HP and Defense (def). Each lumber can buy 160,000 HP directly or 640 def, and each 1 def increases HP by 2%. 2. **Define variables:** Let $x$ be the lumber allocated to HP and $y$ be the lumber allocated to def. We know $x + y = 100$. 3. **Express total HP in terms of $x$ and $y$:** - HP from lumber allocated to HP: $160000x$ - Def from lumber allocated to def: $640y$ - Each def increases HP by 2%, so the HP multiplier is $$1 + 0.02 imes 640y = 1 + 12.8y = 1 + 12.8y$$ - Total HP = HP direct $\times$ multiplier = $$160000x \times (1 + 12.8y)$$ 4. **Rewrite total HP using $y = 100 - x$:** $$\text{Total HP} = 160000x (1 + 12.8 (100 - x)) = 160000x (1 + 1280 - 12.8x) = 160000x (1281 - 12.8x)$$ 5. **Simplify total HP:** $$\text{Total HP} = 160000(1281x - 12.8x^2) = 160000 \times 1281x - 160000 \times 12.8x^2 = 204960000x - 2048000x^2$$ 6. **Maximize total HP:** Set the derivative to zero: $$\frac{d}{dx} (204960000x - 2048000x^2) = 204960000 - 4096000x = 0$$ 7. **Solve for $x$:** $$4096000x = 204960000$$ $$x = \frac{204960000}{4096000} = 50$$ 8. **Interpretation:** To maximize HP, allocate 50 lumber to HP and $100 - 50 = 50$ lumber to def. 9. **Calculate total HP at $x=50$:** $$\text{Total HP} = 160000 \times 50 \times (1 + 12.8 \times 50) = 8{,}000{,}000 \times (1 + 640) = 8{,}000{,}000 \times 641 = 5{,}128{,}000{,}000$$ **Final answer:** Allocate 50 lumber to HP and 50 lumber to def for maximum total HP equal to 5,128,000,000.