The Open Box Problem

The Open Box Problem

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

to be cut off at the corners so that the card can be folded into the

open box. The diagram below shows the sheet of card and the four

corners, which are to be cut off.

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[IMAGE]

There are two objectives I shall be investigating:

1. To determine the size of the four square cuts that will make the

volume of the box as large as possible with any given square sheet

of card.

2. To determine the size of the four square cuts that will make the

volume of the box as large as possible with any given rectangular

piece of card.

First I shall investigate objective 1 as I think it will be easier to

do.

Objective 1: The square sheet of card.

There are 2 ways to solve this problem: I can use algebra, or I can

use trial and improvement. I think I will start by using trial and

improvement; I will construct a series of tables and graphs and see if

I can find any patterns. If so, then I will be able to come up with a

hypothesis, which I can then test to see if I can solve this first

objective. Then I will attempt to solve the same problem using

algebra.

Method 1: Trial and improvement

To do this I need to make up some dimensions and then apply them to

the square. I will choose the dimensions 6x6 to start with. I will

call the length of the square cuttings x. The second drawing shows the

open box with the dimensions included.

[IMAGE]

[IMAGE]

x

[IMAGE][IMAGE] x

[IMAGE][IMAGE][IMAGE][IMAGE] 6cm

[IMAGE] 6-2x

[IMAGE]

[IMAGE]

[IMAGE][IMAGE] 6cm 6-2x x

If we look at the second drawing we can see that to get the two