1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a 9 cm by 9 cm square sheet and folding up the sides. We want to express the volume $V$ as a function of $x$, find the domain of $V(x)$, and determine the dimensions that yield the maximum volume. 2. **Express the volume function:** The length and width of the base after cutting out squares are each $9 - 2x$ cm because we remove $x$ cm from both sides. The height of the box is $x$ cm. Therefore, the volume is: $$V = x(9 - 2x)^2$$ 3. **Find the domain:** Since $x$ is the side length of the cut-out square, it must be positive and less than half the side of the original square to form a box: $$0 < x < \frac{9}{2} = 4.5$$ So the domain is: $$(0,4.5)$$ 4. **Find the dimensions for maximum volume:** From the graph, the maximum volume occurs at $x = 1.5$ cm with $V = 54$ cm³. Calculate length and width: $$\text{Length} = 9 - 2(1.5) = 9 - 3 = 6$$ $$\text{Width} = 9 - 2(1.5) = 6$$ $$\text{Height} = 1.5$$ **Final answers:** - Length = 6 cm - Width = 6 cm - Height = 1.5 cm