The Open Box Problem

Maths Coursework

Jawhari Thomas 11.5

An open box is a box missing 1 of its six surfaces it can be made using either a squared or rectangular sheet of card. Identical squares are cut from the four corners of the card this creates the height of the box it is then folded as shown below. The card is folded along the dotted lines to form the box.

Stage 1 Stage 2 Stage 3

The aim of this exercise is to find the formulae that will enable someone determine the size of the squares cut from the corners of the sheet of card to give the greatest the volume of the box.

I am going to begin the investigation using squares, as this will most probably be easiest. I won’t build the boxes I am going to use simple mathematics to work out the volumes. Firstly I am going to use an example of how I will carry out the experiment using a square 20cm in length. Using this sized length will allow me to only cut off each corner up to 9.9cm as otherwise I will cause me to run out of card. I am going to begin by looking at cutting the squares off as whole numbers. To find the volume of any box we must use the formula:

V = L * W * H

When:

V is Volume

L is Length

W is Width and

H is Height

I know that the both square’s length and width will be equal to 20 but The Length of the box will be equal to

Length – 2 * Cut Size of the Square (e.g. 20 – 2 * Cut Size of the Square) ∴

The volume of the box will have a different formula, which is shown blow

V = L – (2 * Cut Size) * Width – (2 * Cut Size) * Cut Size

I am now going to substitute the cut out size with the sign “∂” Therefore the equation can be changed to:

Volume = 20 – (2∂) * 18* ∂

If I were using a cut out of length 1cm, the equation for this would be as follows:

Volume = 20 – (2 * 1) * 20 - * (2 * 1) * 1

So we can work out through this method that the volume of a box with corners of 1cm² cut out would be:

(20 – 2) * (20 – 2) * 1

18 * 18 * 1

= 324cm³

I used these formulae to construct a spreadsheet in Microsoft Excel, which would allow me to quickly and accurately calculate the volume of the box. Below is the spreadsheet (showing the formulae needed to give the results) and the results for a box of cut size 1.

Here is the table of full whole number cut size for a sheet with a 20cm length and a graph showing the results.

As you can see by the table above, the largest volume is achieved when an area of 3cm² is cut off each corner of the box. I have also drawn a graph to show my results. By looking at This graph, and my table of results, I can see that to achieve the maximum volume I will need to look at cut outs of between 3 and 4 cm² the results of which are presented it a table below:

As you can see by this table, the largest volume is arrived at when the corners cut off the box are 3.3cm². I have also drawn another graph to illustrate these results. This graph shows me that to get even more accurate results, to 2 decimal places, I am going to need to look at cut offs measuring between 3.3 and 3.4 cm. To try to make my results more accurate, I am going to investigate the volume of the box with the cut out to more than 1 decimal place. Below is a table showing the cut out to 2 decimal places, with the largest area achieved highlighted in bold.

The table below shows the cut off measured to 2 decimal places and the half waypoints between the cut sizes. Looking at the table you should be able to see again the largest volume in bold, is with a cut out of 3.335cm. I can see by looking at this graph and also the table that I would need to look between 3.33 and 3.335cm to obtain the maximum volume.

This is the final table for a sheet of length 20cm, showing the maximum area with the cut out measured to 3 decimal places. As you can see ...