1. **State the problem:** We have a box formed by cutting out squares of side $x$ cm from each corner of a $9 \times 9$ 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. The volume formula is: $$V = \text{length} \times \text{width} \times \text{height} = x(9 - 2x)^2$$ 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 using the graph:** The graph shows a maximum volume of 54 at $x = 1.5$. Calculate length and width: $$\text{length} = \text{width} = 9 - 2(1.5) = 9 - 3 = 6$$ Height is $x = 1.5$. 5. **Final answers:** - Volume function: $V = x(9 - 2x)^2$ - Domain: $(0,4.5)$ - Dimensions for maximum volume: - Length = 6 cm - Width = 6 cm - Height = 1.5 cm This means the box with maximum volume is formed by cutting out squares of side 1.5 cm, resulting in a box 6 cm long, 6 cm wide, and 1.5 cm high, with volume 54 cubic centimeters.