An open box is made from a square piece of sheet metal 19 inches on a side by cutting identical squares from the corners and turning up the sides. Express the volume of the box, V, as a function of the length of the side of the square cut from each corner, x.

A. V(x) = 361x
B. V(x) = (19 - 2x)2
C. V(x) = x(19 - 2x)
D. V(x) = x(19 - 2x)2

new length and width = 19-2x

area of base = (19-2x)^2
height = x
so
V = x (19-2x)^2

Well, this box seems to be quite square...literally! To find the volume of the box, we need to consider its dimensions. When we cut identical squares from the corners and turn up the sides, the resulting box will have a length, width, and height.

Now, let's analyze the situation. The original square piece of sheet metal is 19 inches on each side, and we're cutting identical squares from the corners. If we cut a square with side length x from each corner, the dimensions of the resulting box will be (19 - 2x) by (19 - 2x) by x. The reason for 19 - 2x is that we're removing 2x inches from each side.

To find the volume of the box, we multiply these three dimensions together: (19 - 2x) * (19 - 2x) * x.

Expanding this expression, we get V(x) = x(19 - 2x)(19 - 2x). Aha! The volume of the box can be represented by the function V(x) = x(19 - 2x)(19 - 2x), which means the correct answer is option D.

So, the volume of the box can be expressed as V(x) = x(19 - 2x)2. Now that's some creative math for a box.

To find the volume of the open box, we need to determine the dimensions of the box and then calculate the volume using the formula length * width * height.

Let's start by finding the dimensions of the box. When we cut identical squares from the corners of the sheet metal and fold up the sides, we are left with a base (the bottom of the box) that has sides measuring (19 - 2x) inches. The height of the box is simply the length of the square cut from each corner, x inches.

So, the dimensions of the box are (19 - 2x) inches by (19 - 2x) inches by x inches.

Now, we can calculate the volume of the box using the formula V = length * width * height:

V = (19 - 2x) * (19 - 2x) * x
V = x * (19 - 2x) * (19 - 2x)
V = x * (19 - 2x)^2

Therefore, the volume of the open box, V, as a function of the length of the side of the square cut from each corner, x, is:

D. V(x) = x(19 - 2x)^2

To find the volume of the open box, we need to determine the dimensions of the box and then calculate the volume using the formula for the volume of a rectangular prism.

Let's start by understanding the dimensions of the box. The sheet metal was a square with side length 19 inches, and we are cutting identical squares from each corner with side length x. After cutting the squares, the dimensions of the resulting box will be (19 - 2x) inches by (19 - 2x) inches, with x as the height of the box.

Now that we know the dimensions, we can calculate the volume. The volume of a rectangular prism is given by the formula V = lwh, where V is the volume, l is the length, w is the width, and h is the height.

In this case, the length (l) of the box is (19 - 2x) inches, the width (w) is also (19 - 2x) inches, and the height (h) is x inches.

Plugging these values into the volume formula, we get V = (19 - 2x)(19 - 2x)(x).

Simplifying the expression, we have V = (19 - 2x)^2 * x.

So, the correct answer is D. V(x) = x(19 - 2x)^2.