Suppose a box is to be constructed from a square piece of material of side length x bt cutting out a 3-inch square from each corner and turning the sides. Express the volume of the box as a polynomial in the variable x.
since there are two corners cut from each side, the volume is
3(x-6)(x-6)
To find the volume of the box, we need to determine the dimensions of the box first.
When we cut out a 3-inch square from each corner of the original square piece of material, the resulting piece of material that remains will have a length of (x - 2 * 3) inches on each side.
The sides of the box will then be formed by folding up the edges of the remaining material, so the height of the box will be 3 inches.
Therefore, the dimensions of the box will be: length = width = (x - 6) inches, and height = 3 inches.
To find the volume of the box, we multiply the length, width, and height:
Volume = length * width * height
= (x - 6) * (x - 6) * 3
= 3(x - 6)^2
So the polynomial expression for the volume of the box in terms of x is 3(x - 6)^2.