An open box is made from a square prece of cardboard 48 inches on a side by cutting identical squares from the comers and turring up the sides. Use this information to complete parts a through \( c \) Find and interpret \( V(7) \) Select the correct choice below and, if nocessary, fill in the answer box to complete your choice (Simplity your answer Type an integer or a decimal.) A. The volume of the box \( V(7) \) is cardboard is 7 inches B. The volume of the box \( V(7) \) is \( \square \) cubic inches, when the side of the square cut from the center of the pece of of the piece of cardboard is 7 inches the square cut from the center of each edge C. The volume of the box \( V(7) \) is \( \square \) cubic inches, when the side of the square cut from each comer of the piece of cardboard is 7 inches. D. The volume of the box \( V(7) \) is \( \square \) cubic inches, when the side of the square cut from each comer and the center of each edge of the piece of cardboard is 7 inches
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To find \( V(7) \), let’s first visualize the box created from a 48-inch square piece of cardboard. When you cut out squares of 7 inches from each corner and fold up the sides, the height of the box will be 7 inches. The dimensions of the base will be reduced by twice the side of the square cut out. Therefore, each dimension of the base becomes \( 48 - 2(7) = 34 \) inches. Now, you can calculate the volume, \( V \), using the formula for the volume of a box: \( V = \text{length} \times \text{width} \times \text{height} \). In this case, it will be \( V = 34 \times 34 \times 7 \). After calculating, the volume \( V(7) \) equals 8,236 cubic inches. So, the best choice is: C. The volume of the box \( V(7) \) is **8,236** cubic inches, when the side of the square cut from each corner of the piece of cardboard is 7 inches.
