The open box problem

An open box is to be made from a sheet of card.

Identical squares are cut off the four corners of the card as shown below.

The card is then folded along the dotted lines to make the box.

The aim of this project is to determine the size of the square cut out in any given size rectangle sheet of card with the largest volume.

Task 1:

For any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume.

For this task I will use 20cm*20cm square sheet of card. The formula used to get a volume is V=length x Width x height

But if were cutting out 4 small squares out of the 20cm square, will be using this formula= V=length-(2H) x width-(2H) x height

{Height = H, volume = V}

Example: L=20cm, W=20cm, H=1cm

V= H x ((L-2H) x (W-2H))

V= 1 x ((20-2) x (20-2))

= 1 x (18x18)

= 1 x 324 = 324cm3

This calculation would be used to make an investigation on different height. By using different height I will acquire the best size to cut out so that the box has a high volume. The investigation will be processed on excel to give a most accurate results. I will use a graph to differentiate the readings of 20cm2 .

On the graph on the previous page the highest volume is 588cm3 meaning the highest size to cut is 3cm.

Now for a more accurate reading the size ...