Suppose you want to make an open box out of a piece of card board by cutting small squares at the four corners and folding up the sides. if the piece of card board is a square whose sides are 1 m. long, how big a square should you cut from the corners to maximize the volume of the box?

The box of length must be cut from each corner to maximize the volume.

Step-by-step explanation:

It is given that the side of a square card board is 1 m.

Let small squares of length x are cut from the four corners. The dimensions of the box are

Length = (1-2x) m

Width = (1-2x) m

Height = x m

The volume of a cuboid is

where, l is length and b is breadth and h is height.

The volume of a rectangular box.

We need to maximize the volume. Differential above equation with respect to x.

               .... (1)

Equate V'=0 to find the critical points.

Differential equation (1) with respect to x.

At

Volume is maximum at .

At

Volume is minimum at .

Therefore the box of length must be cut from each corner to maximize the volume.