Let's go through each part step by step, providing explanations, details, and modeling as requested. --- ### **a. Modeling the Situation** **i.** *Cutting four equal-sized squares from the corners of an 8.5 by 11-inch sheet of paper.* - **Let** the side length of each square cutout be \( x \) inches. - When you cut a square of side \( x \) from each corner, you remove \( x \) from both the length and the width, twice (from both ends). So, after cutting, the dimensions of the base (when the sides are folded up): - **Length of base:** \( 11 - 2x \) - **Width of base:** \( 8.5 - 2x \) - **Height of box:** \( x \) **ii.** *Folding up the sides and taping them together at the edges.* - After cutting, fold up the flaps (sides) to form the sides of the box. - The height of the box will be the same as the side length of the square that was cut out (\( x \)). --- ### **b. Do the cutouts have to be square? Explain.** - **No, the cutouts do not have to be square.** - However, to ensure that the box has vertical sides and the base is rectangular, the cutouts **should** be square. Otherwise, the sides won’t meet neatly at the corners and the base will not be a rectangle. - **In practice,** to get a box with vertical sides and a rectangular base, squares are cut from each corner. If non-square rectangles are cut, the sides will not match up properly. --- ### **c. What quantities in this situation vary? What quantities are constant?** - **Variables (Vary):** - The side length of the squares cut from each corner (\( x \)) - The resulting height of the box (\( x \)) - The resulting length and width of the box (\( 11-2x \) and \( 8.5-2x \)) - The volume of the box (\( V \)) - **Constants (Do not vary):** - The original length of the paper (11 in) - The original width of the paper (8.5 in) --- ### **d. Describe how the configuration of the box changes as \( x \) increases. What is the largest value that makes sense for \( x \)?** - **As \( x \) increases:** - The height of the box increases. - The length and width of the base decrease (\( 11-2x \), \( 8.5-2x \)); the base gets smaller. - At very small \( x \), the box is shallow and has a large base. - At very large \( x \), the box is tall but the base becomes very small. - **Largest value that makes sense for \( x \):** - \( x \) must be less than half of the shorter side, otherwise you’d be cutting away the whole side. - The shorter side is 8.5 in, so \( x < 4.25 \) inches. - **So, \( 0 < x < 4.25 \) inches.** --- ### **e. Volume Variation as \( x \) Varies from 0 to 4.25** **Volume formula:** \[ V(x) = x \cdot (11 - 2x) \cdot (8.5 - 2x) \] - As \( x \) increases from 0: - At \( x = 0 \): Volume is 0 (no box). - As \( x \) increases, volume increases, reaches a maximum, then decreases back to 0 as \( x \to 4.25 \). - At \( x = 4.25 \): either length or width becomes zero, so volume is again 0. --- ### **f. Sketch a Graph: Volume vs. Cutout Length** **Axes:** - **x-axis:** Side length of square cutout (\( x \)), from 0 to 4.25 inches - **y-axis:** Box volume (\( V(x) \)), in cubic inches **Graph shape:** - Starts at (0,0) - Rises to a maximum (somewhere between 1 and 2 inches) - Falls back to (4.25, 0) ``` | | /\ | / \ | / \ |____/______________\____ 0 x_max 4.25 ``` **Label x-axis:** "Cutout side length (inches)" **Label y-axis:** "Volume (cubic inches)" --- ### **g. Measure the Length of Your Cutout (with a ruler)** **You need to measure the side of the square you actually cut out.** - Let’s suppose you measure and find: **Answer:** \( x = 1.5 \) inches --- ### **h. Compute Box Dimensions and Volume for Your Box** Given \( x = 1.5 \) inches: **Height:** \( x = 1.5 \) in **Length of box’s base:** \( 11 - 2x = 11 - 3 = 8 \) in **Width of box’s base:** \( 8.5 - 2x = 8.5 - 3 = 5.5 \) in **Volume:** \[ V = \text{height} \times \text{length} \times \text{width} \] \[ V = 1.5 \times 8 \times 5.5 = 1.5 \times 44 = 66~\text{cubic inches} \] --- ### **Summary Table for Your Box** | Quantity | Value | |------------------------|------------| | Height | 1.5 in | | Length of base | 8 in | | Width of base | 5.5 in | | Volume | 66 in³ | --- **If you have a different value for \( x \), just plug it into the formulas:** - Height = \( x \) - Length = \( 11 - 2x \) - Width = \( 8.5 - 2x \) - Volume = \( x \cdot (11-2x) \cdot (8.5-2x) \) --- **Let me know if you’d like to see the graph plotted or need help with a different value for your measured cutout!**

a. Modeling the situation: i. Cutting four squares of side \( x \) inches from each corner results in a base with dimensions: - Length: \( 11 - 2x \) inches - Width: \( 8.5 - 2x \) inches - Height: \( x \) inches ii. Folding up the sides forms a box with these dimensions, with the folds creating the height. b. Do the cutouts have to be square? No, they do not have to be square. However, squares are used to ensure the sides are vertical and the box is rectangular. c. Varying and constant quantities: - Vary: \( x \) (size of cutouts), box volume, base length, base width, box height - Constant: original sheet dimensions (11" and 8.5") d. As \( x \) increases: - The height increases. - The base dimensions decrease. - Largest \( x \) is less than half the shorter side: \( x < 4.25 \) inches. e. Volume variation: \[ V(x) = x (11 - 2x)(8.5 - 2x) \] Volume increases from 0 at \( x=0 \), reaches a maximum, then decreases back to 0 at \( x=4.25 \). f. Graph description: - x-axis: \( x \) from 0 to 4.25 inches - y-axis: \( V(x) \) - Shape: upward curve from 0, peak at maximum volume, downward back to 0 at \( x=4.25 \). g. Measured cutout side length: (Example answer) \( x = 1.5 \) inches h. For \( x=1.5 \) inches: - Height: 1.5 in - Length: \( 11 - 2(1.5) = 8 \) in - Width: \( 8.5 - 2(1.5) = 5.5 \) in - Volume: \( 1.5 \times 8 \times 5.5 = 66 \text{ in}^3 \) --- **Final summarized answers:** **a.** - Base dimensions: length = \(11 - 2x\), width = \(8.5 - 2x\), height = \(x\). **b.** - No, squares are not necessary, but they ensure proper box shape. **c.** - Varying: \( x \), volume, base dimensions. - Constant: sheet dimensions (11", 8.5"). **d.** - The box height increases with \( x \), base decreases; maximum \( x \) is just under 4.25". **e.** - \( V(x) = x(11 - 2x)(8.5 - 2x) \), peaks at some \( x \) between 0 and 4.25. **f.** - Graph: volume curve starting at 0, rising to a maximum, then declining back to 0 at \( x=4.25 \). **g.** - Example measurement: 1.5 inches. **h.** - Height: 1.5 in - Length: 8 in - Width: 5.5 in - Volume: 66 in³