Open Box Problem

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In this investigation, a box without a lid must be made from a sheet of card, as shown below.

Identical squares must be cut out of each corner and the dotted lines folded along to form the sides of the box.

The goal of the investigation is to find out a relationship between the size of the initial piece of card, the size of the identical corner squares and the volume of the resulting open box. This will allow me to say what size corner square will produce the box of the largest volume, for any given rectangular sheet of card.

To begin with, I am going to investigate the size of the corner square that must be cut out to make an open box of the largest volume, for any sized square sheet of card. Once I have found the formula that allows me to find this out easily, I will progress to using an initial piece of card that is rectangular in shape.

The formula used to obtain the volume of a box is

VOLUME = Length * Width * Height

(where * is multiplication)

To show a simple example of how this formula works with the open box, I will first of all use a initial piece of card that is 20cm by 20 cm, and a corner square (from now on called 'cut-off') of 2cm by 2cm.

Once the cut-offs are taken away, the net will look like this.

From this we can see that when the dotted lines are folded along, there is a height of 2cm, a length of 16cm and a width of 16cm.

Since Volume = Length * Width * Height, the volume of this open box is 16*16*2 = 512cm3

We can also see relationships between the cut-off and the dimensions of the net from this example.

. The size of the cut-off is the same size as the height of the box

2. The initial sizes of the length and width have decreased by 2 times the size of the cut-off

So, when the initial piece of card is square, the formula for the volume can be written:

Volume = (Length-2*Cut-off) * (Width-2*Cut-off) * Cut-off
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V = (L-2C) * (W-2C) * C

V = (20-2*2) * (20-2*2) * 2

V = 16*16*2

V= 512cm3

And since the length and width are always equal to each other in a square, the formula can be simplified:

Volume = (Side-2*Cut-off)2 * Cut-off

V = (S-2C)2 *C

(where S is either the length or the width)

Using this theory I created a spreadsheet that allowed me to make tables of results quickly and efficiently.

This spreadsheet is set up so that when the side length and cut-off are ...

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