Final answer:
To calculate the volume of the open-top box made from cardboard, subtract twice the side length of the corner squares (x) from each cardboard dimension and multiply length, width, and height. The volume formula is V = (13 - 2x)(14 - 2x)(x).
Step-by-step explanation:
Finding the Formula for Volume of an Open-Top Box
To find a formula for the volume of an open-top box made from a piece of cardboard measuring 13 inches by 14 inches, we start by understanding that cutting out squares of side length x from each corner and folding up the sides will change the dimensions of the cardboard for the base of the box. Once the squares are cut, the new length and width of the box will be (13-2x) inches and (14-2x) inches, respectively, as x is removed from each side.
The height of the box is simply x, as this is the size of the square that we fold up. Therefore, the volume V of the box is given by the product of its length, width, and height. The formula for the volume of the box V is:
V = (length - 2x)(width - 2x)(height)
V = (13 - 2x)(14 - 2x)(x)
This is the required formula for the volume of the open-top box in terms of x, where x must be less than 6.5 inches to ensure that the squares cut out don't overlap and the resulting dimensions are all positive.