A manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squares from the corners and folding up the sides. The Research and Development Department asks you to determine the size of the square that produces a box of greatest volume. Proceed as follows. Let x be the length of a side of the square to be cut and V be the volume of the resulting box. Show that V = x(6-2x)^2. I don't understand how I'm suppose to show what they are asking.

make a sketch of a 6 by 6 square.

Draw squares at each corner of x by x
(You could actually cut them out with scissors)
This would allow you to fold up the sides to form a box
Wouldn't each side of the base remaining be 6 - 2x, since x units were taken away at each end ?
Wouldn't the height of the box be x ?

volume = base x height
= (6-2x)(6-2x(x)
= x(6-2x)^2

I just wasn't sure if the answer was simply a sketch, 6 - 2x each side with the cut-outs. Thanks!!

To show that V = x(6-2x)^2, we need to understand the process of constructing the box and how it relates to the equation.

First, imagine a square-shaped sheet of cardboard with side length x. The manufacturer cuts small squares with side length x from each corner. This will leave a rectangular piece in the middle, which will be used to create the sides of the box.

Now, let's examine one side of the box. When the sides are folded up, the height of the box will be the length of the small square cut from the corner (which is also x), and the width will be the length of the rectangular piece, which is 6 - 2x (since we subtract 2x from each side of the original square).

The length of the box will also be 6 - 2x because the original side length of the cardboard is x, and we cut x from each corner, leaving 6 - 2x.

Therefore, the volume of the box can be determined by multiplying the length, width, and height of the box:

V = (6 - 2x) * x * x
V = x(6 - 2x)^2

So, by showing that the volume of the box can be expressed as V = x(6-2x)^2, we have precisely expressed the relationship between the dimensions of the box and its volume.