Question

Suppose a box is to be constructed from a square piece of material of side length x by cutting out a 2-inch square from each corner and turning up the sides. express the volume of the box as a polynomial in the variable x

Answer 1

Consider the attached figure. The central square is the "floor" of the box, and it is a (x-2)-by-(x-2) square.

The four rectangles of the side will be the "walls" of the box, and they are (x-2)-by-2 rectangles.

The volume is given by the base area multiplied by the height of the walls, so we have

$\underbrace{(x-2)(x-2)}_{\text{base area}}\times 2 = 2(x^2-4x+4)= 2x^2-8x+8$