A cardboard manufacturer wishes to make open boxes from square pieces of cardboard of side 12 in. by cutting equal squares from the four corners and turning up the sides. Let x inches be the length of the side of the the square to be cut out. Express the Volume (V) of the box (in cubic inches) as a function of x.

The number of inches in the dimension of the box are x, 12 - 2x and 12 - 2x.

V (x) = 144x - 48x2 + 4x3

v = x(12-2x)^2

Well, well, well, we've got ourselves a cardboard conundrum here! Alright, let's get slicing and dicing.

First, let's visualize this operation. We have a square piece of cardboard with side length 12 inches. We're going to cut out equal squares from all four corners and then fold up the sides to create an open box.

Now, when we cut out squares of side length x from each corner, we're essentially reducing the dimensions of the original cardboard. The new dimensions will be (12 - 2x) inches for the length and (12 - 2x) inches for the width.

To find the volume of the box, we multiply the length, width, and height. The height here is x inches because that's the amount we'll fold up the sides.

So, the volume (V) of the box can be expressed as:

V = (12 - 2x) * (12 - 2x) * x

And there you have it! The volume of the box as a function of x.

To find the volume of the open box created from cutting squares out of the corners of a 12-inch square piece, we need to determine the dimensions of the resulting box.

Let's start by drawing a diagram representing the initial 12-inch square piece of cardboard and the squares to be cut out from each corner:

```
+---x---+---------------------------+
| | |
| | |
x | |
| | |
| | |
+-----------------------------------+
```

Here, x represents the length of the side of the square to be cut out.

After cutting out the squares and folding up the sides, the dimensions of the resulting box will be as follows:

```
+-----------------------------------+
| |
| +---------------------+ |
| | | |
| | | |
| | | |
| +---------------------+ |
| |
+-----------------------------------+
```

The base of the box (highlighted with `+---+`) will have dimensions of (12 - 2x) inches by (12 - 2x) inches. The height of the box will be equal to x inches.

Now, we can calculate the volume of the box, V, by multiplying its length, width, and height:

V = (12 - 2x) * (12 - 2x) * x

Simplifying this expression gives us the final function for the volume of the box as a function of x:

V(x) = x(144 - 48x + 4x^2)

Therefore, the volume of the open box is given by the function V(x) = x(144 - 48x + 4x^2) in cubic inches.

A cardboard box manufacturer wishes to make open boxes from square pieces of

cardboard of side 12 inches by cutting equal squares from the four corners and turning
up the sides. Let x inches be the length of the sides of the square to be cut out. Express
the number of cubic inches in the volume of the box as a function of x.