An open box is to be made from a rectangular piece of material by cutting equal squares of length x from each corner and turning up the sides. If the material is 18 inches long and 12 inches wide. What is the volume of the box as a function of x?

well, x is cut off both sides, and x is the box's height, so

v(x) = (18-2x)(12-2x)(x)

To find the volume of the box as a function of x, we need to consider the dimensions of the box after the corners are cut and the sides are turned up.

Let's break it down step by step:

1. Start with a rectangular piece of material with length = 18 inches and width = 12 inches.

2. Cut equal squares of length x from each corner. This will reduce the length and width of the rectangular piece by 2x. So the new length will be 18 - 2x inches, and the new width will be 12 - 2x inches.

3. Turn up the sides to form an open box. When the sides are turned up, the height of the box will be x inches.

Now we have the dimensions of the open box: length = 18 - 2x inches, width = 12 - 2x inches, and height = x inches.

To find the volume of the box, multiply the length, width, and height:

Volume = (18 - 2x)(12 - 2x)(x)

Simplifying this expression further is possible, but since we only need the volume of the box as a function of x, the equation above provides a suitable answer.