An open box is made from a square piece of cardboard 48 inches on a side by cutting identical squares from the comers and turning up the sides. Use this information to complete parts a through \( c \) Find and interpret \( V(8) \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice (Simplify your answer. Type an integer or a decimal.) A. The volume of the box \( V(8) \) is cubic inches, when the side of the square cut from the center of each edge of the piece of cardboard is 8 inches B. The volume of the box \( V(8) \) is cubic inches, when the side of the square cut from the center of the piece of cardboard is 8 inches. C. The volume of the box \( V(8) \) is cubic inches, when the side of the square cut from each comer of the piece of cardboard is 8 inches D. The volume of the box \( V(8) \) is cubic inches, when the side of the square cut from each comer and the center of each edge of the piece of cardboard is 8 inches

To find the volume of the box when 8-inch squares are cut from each corner of the cardboard, first, let's determine the dimensions of the box after the cuts. The original cardboard is 48 inches on each side. Cutting out 8-inch squares from each corner reduces each dimension by \(8 + 8 = 16\) inches. This means the new dimensions of the box are \(48 - 16 = 32\) inches for both the length and the width, while the height will be equal to the side length of the corner cutouts, which is 8 inches. Using the volume formula \(V = \text{length} \times \text{width} \times \text{height}\), we calculate: \[ V(8) = 32 \times 32 \times 8 = 8192 \text{ cubic inches} \] So, the correct choice is C: The volume of the box \( V(8) \) is 8192 cubic inches when the side of the square cut from each corner of the piece of cardboard is 8 inches. Now, just when you're thinking this is all there is to it, remember that volume in real-life applications helps us figure out how much stuff can fit into a container. So, that 8192 cubic inches might just mean you can store that perfect stack of holiday decorations, or maybe even a sizeable collection of board games! Lastly, when solving any geometry problem like this, always double-check the dimensions you derive after cutting! It's very easy to mix up which lengths and widths you've adjusted, leading to a potential headache down the line (not to mention an incorrect volume!).