1. **Problem Statement:** We want to find the volume $V(x)$ of an open-top box formed by cutting out squares of side length $x$ from each corner of a $12 \times 12$ inch square sheet and folding up the sides. 2. **Volume Formula:** The volume of the box is given by the product of height, length, and width: $$V = h \times l \times w$$ Here, height $h = x$, length $l = 12 - 2x$, and width $w = 12 - 2x$. 3. **Expressing Volume as a Function of $x$:** $$V(x) = x(12 - 2x)^2$$ Expanding: $$V(x) = x(144 - 48x + 4x^2) = 144x - 48x^2 + 4x^3$$ 4. **Finding Critical Points:** To find the maximum volume, differentiate $V(x)$ with respect to $x$: $$\frac{dV}{dx} = 144 - 96x + 12x^2$$ Factor out 12: $$\frac{dV}{dx} = 12(12 - 8x + x^2) = 12(2 - x)(6 - x)$$ 5. **Solve for Critical Points:** Set derivative equal to zero: $$12(2 - x)(6 - x) = 0$$ Critical points are at: $$x = 2 \quad \text{and} \quad x = 6$$ 6. **Evaluate Volume at Critical Points and Endpoints:** - At $x=2$: $$V(2) = 2(12 - 4)^2 = 2 \times 8^2 = 2 \times 64 = 128$$ - At $x=0$ (no cut): $$V(0) = 0$$ - At $x=6$ (cuts remove entire length): $$V(6) = 6(12 - 12)^2 = 6 \times 0 = 0$$ 7. **Interpretation:** The maximum volume of the box is $128$ cubic inches when the cut-out squares have side length $x=2$ inches. This matches the physical intuition that cutting too small or too large squares results in zero or small volume.