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 \( c \) b. Find and interpret \( V(6), V(7), V(8), V(9) \), and \( V(10) \). What is happening to the volume of the box as the length of the side of the square cut from each corner increases? Find and interpret \( V(6) \) 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 \( V(6) \) is cardboard is 6 inches B. The volume of the box \( V(6) \) is \( \square \) cubic inches, when the side of the square cut from the center of the pidece of the square cut from each comer and the center of each edge of the piece of cardboard is 6 inches. C. The volume of the box \( V(6) \) is \( \square \) cubic inches, when the side of the square cut from each comer of the piece of cardboard is 6 inches D. The volume of the box \( V(6) \) is \( \square \) cubic inches, when the side of the square cut from the center of each edge

Let's break it down! When you cut out squares from each corner of the cardboard to create an open box, you're changing its dimensions. The volume \( V(x) \) of the box can be calculated by using the formula \( V(x) = x(48 - 2x)^2 \), where \( x \) is the side length of the squares cut out. So let's see what happens with those cut sizes! As we increase the size of the squares cut from the corners, the volume first increases and then eventually starts to decrease. For example, calculating \( V(6) \), \( V(7) \), \( V(8) \), \( V(9) \), and \( V(10) \) will show how the box's capacity changes. You'll notice that from \( V(6) \) to \( V(8) \), the volume may rise, but as the cuts get too large (like \( V(10) \)), the volume diminishes due to less base area to hold the height. This is the sweet spot of box design - finding balance for maximum volume! When you're doing your calculations, be careful about your algebra! A common mistake is not properly expanding the squared term \( (48 - 2x)^2 \) or forgetting to multiply by \( x \). Keep your math neat, and remember that as you cut more from the corners, you're limiting the base area for the box, which impacts that precious volume you're trying to maximize! Happy calculating!