An open box is made from a square piece of cardboard 48 inches on a side by cutting identical squares from the comers and tuming up the sides. Use this information to complete parts a through c . a. Express the volume of the box, V , as a function of the length of the side of the square cut from each comer, x . \( \mathrm{V}(\mathrm{x})=\square \) (Simplify your answer.)

To express the volume \( V \) of the box as a function of \( x \), we start with a square piece of cardboard measuring 48 inches on each side. After cutting out squares of side length \( x \) from each corner, the new dimensions of the box will be reduced by \( 2x \) in both length and width. Therefore, the length and width of the base of the box become \( (48 - 2x) \). The height of the box will simply be \( x \). The volume \( V \) of the box can be calculated using the formula for the volume of a rectangular prism: \[ V(x) = \text{length} \times \text{width} \times \text{height} = (48 - 2x)(48 - 2x)(x) \] So, simplifying this further: \[ V(x) = x(48 - 2x)^2 \] Which expands to: \[ V(x) = x(2304 - 192x + 4x^2) = 2304x - 192x^2 + 4x^3 \] Thus, \[ V(x) = 4x^3 - 192x^2 + 2304x \] Now the volume is neatly expressed as a cubic function in terms of \( x \)!