1. **Problem Statement:** We have a square sheet of metal 12 inches on each side. Squares of side length $x$ inches are cut from each corner, and the sheet is folded to form an open box. 2. **Expressing the Volume $V$ as a function of $x$:** - The original sheet is $12 \times 12$ inches. - After cutting squares of side $x$ from each corner, the new length and width of the base are each $12 - 2x$ inches (since squares are cut from both sides). - The height of the box is $x$ inches (the side length of the cut square). 3. **Volume formula:** $$V(x) = \text{length} \times \text{width} \times \text{height} = (12 - 2x)(12 - 2x)(x) = x(12 - 2x)^2$$ 4. **Domain of $V$:** - $x$ must be positive: $x > 0$ (since the cut squares have positive side length). - The length and width must be positive: $12 - 2x > 0 \Rightarrow x < 6$. 5. **Final domain:** $$0 < x < 6$$ **Answer:** $$V(x) = x(12 - 2x)^2, \quad 0 < x < 6$$